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Mathematica 







c 



RUSSIA 



VOLUME I 



CAMBRIDGE UNIVERSITY PRESS 



$4J.oo the set of 3 volumes 



PRINCIPIA MATHEMATICA 

BY 

A.N. WHITEHEAD 

AND 

BERTRAND RUSSELL 

Principia Mathematica was first published 
in 19 1 0—13 ; this is the fifth impression of 
the second edition of 192^-7. 

The Principia has long been recognized 
as one of the intellectual landmarks of the 
century. It was the first book to show 
clearly the close relationship between 
mathematics and formal logic. Starting 
from a minimal number of axioms, White- 
head and Russell display the structure of 
both kinds of thought. No other book 
has had such an influence on the subse- 
quent history of mathematical philosophy. 





£12. 125. net the set of 3 volumes 



PRINCIPIA MATHEMATICA 

BY 

ALFRED NORTH WHITEHEAD 

AND 

BERTRAND RUSSELL, F.R.S. 



VOLUME I 



SECOND EDITION 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 

1963 



PUBLISHED BY 

THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS 

Bentley House, 200 Euston Road, London, N.W. 1 

American Branch : 32 East 57th Street, New York 22, N.Y. 

West African Office: P.O. Box 83, Ibadan, Nigeria 



First Edition 1910 

Second Edition 1927 

Reprinted 1950 

1957 

1960 

1963 



PRESTON 



v>ri> 



^4i 3<^ y 



Jteik 



'tf(^ 



84 



F»re< printed in Great Britain at the University Press, Cambridge 

Reprinted by offset-Mho 

by Messrs Lowe & Brydone (Printers) Ltd., London, N.W. 10 



PREFACE 

THE mathematical treatment of the principles of mathematics, which is 
the subject of the present work, has arisen from the conjunction of two 
different studies, both in the main very modern. On the one hand we have 
the work of analysts and geometers, in the way of formulating and systematising 
their axioms, and the work of Cantor and others on such matters as the theory 
of aggregates. On the other hand we have symbolic logic, which, after a 
necessary period of growth, has now, thanks to Peano and his followers, 
acquired the technical adaptability and the logical comprehensiveness that are 
essential to a mathematical instrument for dealing with what have hitherto 
been the beginnings of mathematics. From the combination of these two 
studies two results emerge, namely (1) that what were formerly taken, tacitly 
or explicitly, as axioms, are either unnecessary or demonstrable; (2) that the 
same methods by which supposed axioms are demonstrated will give valuable 
results in regions, such as infinite number, which had formerly been regarded 
as inaccessible to human knowledge. Hence the scope of mathematics is 
enlarged both by the addition of new subjects and by a backward extension 
into provinces hitherto abandoned to philosophy. 

The present work was originally intended by us to be comprised in a 
second volume of The Principles of Mathematics. With that object in view, 
the writing of it was begun in 1900. But as we advanced, it became in- 
creasingly evident that the subject is a very much larger one than we had 
supposed; moreover on many fundamental questions which had been left 
obscure and doubtful in the former work, we have now arrived at what we 
believe to be satisfactory solutions. It therefore became necessary to make 
our book independent of The Principles of Mathematics. We have, however, 
avoided both controversy and general philosophy, and made our statements 
dogmatic in form. The justification for this is that the chief reason in favour 
of any theory on the principles of mathematics must always be inductive, 
i.e. it must lie in the fact that the theory in question enables us to deduce 
ordinary mathematics. In mathematics, the greatest degree of self-evidence 
is usually not to be found quite at the beginning, but at some later point; 
hence the early deductions, until they reach this point, give reasons rather' 
for believing the premisses because true consequences follow from them, than 
for believing the consequences because they follow from the premisses. 

In constructing a deductive system such as that contained in the present 
work, there are two opposite tasks which have to be concurrently performed. 
On the one hand, we have to analyse existing mathematics, with a view 
to discovering what premisses are employed, whether these premisses are 
mutually consistent, and whether they are capable of reduction to more 
fundamental premisses. On the other hand, when we have decided upon oUr 
premisses, we have to build up again as much as may seem necessary of the 
data previously analysed, anji as many other consequences of our premisses 
as are of sufficient general interest to deserve statement. The preliminary 
labour of analysis does not appear in the final presentation, which merely 
sets forth the outcome of the analysis in certain undefined ideas and 



VI PREFACE 

undemonstrated propositions. It is not claimed that the analysis could not 
have been carried farther: we have no reason to suppose that it is impossible 
to find simpler ideas and axioms by means of which those with which we 
start could be defined and demonstrated. All that is affirmed is that the 
ideas and axioms with which we start are sufficient, not that they are 
necessary. 

In making deductions from our premisses, we have considered it essential 
to carry them up to the point where we have proved as much as is true in 
whatever would ordinarily be taken for granted. But we have not thought 
it desirable to limit ourselves too strictly to this task. It is customary to 
consider only particular cases, even when, with our apparatus, it is just as 
easy to deal with the general case. For example, cardinal arithmetic is 
usually conceived in connection with finite numbers, but its general laws hold 
equally for infinite numbers, and are most easily proved without any mention 
of the distinction between finite and infinite. Again, many of the properties 
commonly associated with series hold of arrangements which are not strictly 
serial,, but have only some of the distinguishing properties of serial arrange- 
ments. In such cases, it is a defect in logical style to prove for a particular 
class of arrangements what might just as well have been proved more generally. 
An analogous process of generalization is involved, to a greater or less degree, 
in all our work. We have sought always the most general reasonably simple 
hypothesis from which any given conclusion could be reached. For this reason, 
especially in the later parts of the book, the importance of a proposition 
usually lies in its hypothesis. The conclusion will often be something which, 
in a certain class of cases, is familiar, but the hypothesis will, whenever possible, 
be wide enough to admit many cases besides those in which the conclusion is 
familiar. 

We have found it necessary to give very full proofs, because otherwise it 
is scarcely possible to see what hypotheses are really required, or whether our 
results follow from our explicit premisses. (It must be remembered that we 
are not affirming merely that such and such propositions are true, but also 
that the axioms stated by us are sufficient to prove them.) At the same time, 
though full proofs are necessary for the avoidance of errors, and for convincing 
those who may feel doubtful as to our correctness, yet the" proofs of propo- 
sitions may usually be omitted by a reader who is not specially interested in 
that part of the subject concerned, and who feels no doubt of our substantial 
accuracy on the matter in hand. The reader who is specially interested in 
some particular portion of the book will probably find it sufficient, as regards 
earlier portions, to read the summaries of previous parts, sections, and 
numbers, since these give explanations of the ideas involved and statements of 
the principal propositions proved. The proofs in Part I, Section A, however, 
are necessary, since in the course of them the maimer of stating proofs is 
explained. The proofs of the earliest propositions are given without the 
omission of any step, but as the work proceeds the proofs are gradually 
compressed, retaining however sufficient detail to enable the reader by the 
help of the references to reconstruct proofs in which no step is omitted. 

The order adopted is to some extent optional. For example, we have treated 
cardinal arithmetic and relation-arithmetic before series, but we might have 
treated series first. To a great extent, however, the order is determined by 
logical necessities. 



PREFACE VU 

A very large part of the labour involved in writing the present work has 
been expended on the contradictions and paradoxes which have infected logic 
and the theory of aggregates. We have examined a great number of hypo- 
theses for dealing with these contradictions ; many such hypotheses have been 
. advanced by others, and about as many have been invented by ourselves. 
Sometimes it has cost us several months' work to convince ourselves that 
a hypothesis was untenable. In the course of such a prolonged study, we 
have been led, as was to be expected, to modify our views from time to time ; 
but it gradually became evident to us that some form of the doctrine of types 
must be adopted if the contradictions were to be avoided. The particular 
form of the doctrine of types advocated in the present work is not logically 
indispensable, and there are various other forms equally compatible with the 
truth of our deductions. We have particularized, both because the form of 
the doctrine which we advocate appears to us the most probable, and because 
it was necessary to give at least one perfectly definite theory which avoids 
the contradictions. But hardly anything in our book would be changed by the 
adoption of a different form of the doctrine of types. In fact, we may go 
farther, and say that, supposing some other way of avoiding the contradictions 
to exist, not very much of our book, except what explicitly deals with types, 
is dependent upon the adoption of the doctrine of types in any form, so soon 
as it has been shown (as we claim that we have shown) that it is possible 
to construct a mathematical logic which does not lead to contradictions. It 
should be observed that the whole effect of the doctrine of types is negative : 
it forbids certain inferences which would otherwise be valid, but does not 
permit any which would otherwise be invalid. Hence we may reasonably 
expect that the inferences which the doctrine of types permits would remain 
valid even if the doctrine should be found to be invalid. 

Our logical system is wholly contained in the numbered propositions, which 
are independent of the Introduction and the Summaries. The Introduction 
and the Summaries are wholly explanatory, and form no part of the chain of 
deductions. The explanation of the hierarchy of types in the Introduction 
differs slightly from that given in #12 of the body of the work. The latter 
explanation is stricter and is that which is assumed throughout the rest of 
the book. 

The symbolic form of the work has been forced upon us by necessity : 
without its help we should have been unable to perform the requisite 
reasoning. It has been developed as the result of actual practice, and is not 
an excrescence introduced for the mere purpose of exposition. The general 
method which guides our handling of logical symbols is due to Peano. His 
great merit consists not so much in his definite logical discoveries nor in the 
details of his notations (excellent as both are), as in the fact that he first 
showed how symbolic logic was to be freed from its undue obsession with the 
forms of ordinary algebra, and thereby made it a suitable instrument for 
research. Guided by our study of his methods, we have used great freedom 
in constructing, or reconstructing, a symbolism which shall be adequate to 
deal with all parts of the subject. No symbol has been introduced except 
on the ground of its practical utility for the immediate purposes of our 
reasoning. 

A certain number of forward references will be found in the notes and 
explanations. Although we have taken every reasonable precaution to secure 



Vlll PREFACE 

the accuracy of these forward references, we cannot of course guarantee their 
accuracy with the same confidence as is possible in the case of backward 
references. 

Detailed acknowledgments of obligations to previous writers have not very 
often been possible, as we have had to transform whatever we have borrowed, 
in order to adapt it to our system and our notation. Our chief obligations 
will be obvious to every reader who is familiar with the literature of the 
subject. In the matter of notation, we have as far as possible followed Peano, 
supplementing his notation, when necessary, by that of Frege or by that of 
Schroder. A great deal of the symbolism, however, has had to be new, not 
so much through dissatisfaction with the symbolism of others, as through the 
fact that we deal with ideas not previously symbolised. In all questions of 
logical analysis, our chief debt is to Frege. Where we differ from him, it is 
largely because the contradictions showed that he, in common with all other 
logicians ancient and modern, had allowed some error to creep into his pre- 
misses; but apart from the contradictions, it would have been almost impossible 
to detect this error. In Arithmetic and the theory of series, our whole work 
is based on that of Georg Cantor. In Geometry we have had continually 
before us the Writings of V. Staudt, Pasch, Peano, Pieri, and Veblen. 

We have derived assistance at various stages from the criticisms of friends, 
notably Mr G, G. Berry of the Bodleian Library and Mr R. G. Hawtrey. 

We have to thank the Council of the Royal Society for a grant towards the 
expenses of printing of £200 from the Government Publication Fund, and also 
the Syndics of the University Press who have liberally undertaken the greater 
portion of the expense incurred in the production of the work. The technical 
excellence, in all departments, of the University Press, and the zeal and courtesy 
of its officials, have materially lightened the task of proof-correction. 

The second volume is already in the press, and both it and the third will 
appear as soon as the printing can be completed. 

A. N. W. 

B. R. 

Cambridge, 

November, 1910. 



CONTENTS OF YOLUME I 



PREFACE 



ALPHABETICAL LIST OF PROPOSITIONS REFERRED TO BY 

NAMES . 

INTRODUCTION TO THE SECOND EDITION . . . 

INTRODUCTION . . . . •. ' . . 

Chapter I. Preliminary Explanations of Ideas and Notations 
Chapter II. Th« Theory of Logical Types . . 
Chapter III. Incomplete Symbols . . . . . . 

PART I. MATHEMATICAL LOGIC 

Summary of Part I . .. . . . .... 

Section A. The Theory of Deduction 

#1. Primitive Ideas and Propositions 

#2. Immediate Consequences o£ the Primitive Propositions 

#3. The Logical Product of two Propositions .... 

#4.. Equivalence and Formal Rules ...... 

#& Miscellaneous Propositions . . . . 

Section B. Theory of Apparent Variables ..... 

#9. Extension of the Theory of Deduction from Lower to Higher 
Types of Propositions . . . . . . 

#10. Theory of Propositions containing one Apparent Variable 

•11. Theory of two Apparent Variables ..... 

#12. The Hierarchy of Types and the Axiom of Reducibility 

#13. Identity . . 

#14. Descriptions- . . . 

Section C. Classes and Relations . 

#20.- General Theory of Classes . 

#21. General Theory of Relations 

#22. Calculus of Classes 

#23. Calculus of Relations . 

#24. The Universal Glass* the Null Class, 

Classes ..... 
#25; The Universal Relation, the Null Relation, and the Existence 

of Relations . . . 



and the Existence of 



PAGE 

v 

xii 

xiii 

1 

4 

37 

66 

87 

90 

91 

98 

109 

115 

123 

127 

127 
138 
151 
161 
168 
173 

187 

187 
200 
205 
213 

216 

228 



CONTENTS 



Section 
#30. 
*31. 
#32. 

#33. 
•34. 
•35. 
#36. 
#37. 
#38. 



Section 
#40. 
#41. 
<#42. 
#43. 



D. Logic op Relations 
Descriptive Functions . 
Converses of Relations 
Referents and Relata of a given Term with respect to a given 

Relation . . . . . . . 

Domains, Converse Domains, and Fields of Relations . 

The Relative Product of two Relations . . 

Relations with Limited Domains and Converse Domains 

Relations with Limited Fields 

Plural Descriptive Functions 

Relations and Classes derived from a Double Descriptive 

Function . . . . 
Note to Section D . . . 

E. Products and Sums of Classes 
Products and Sums of Classes of Classes 
The Product and Sum of a Class of Relations 
Miscellaneous Propositions . . . . . 
The Relations of a Relative Product to its Factors 



PART II. PROLEGOMENA TO CARDINAL ARITHMETIC 
Summary of Part II . . . 

Section A. Unit Classes and Couples . 
#50. Identity and Diversity as Relations 
#51. Unit Classes 
#52. The Cardinal Number 1 

#53. Miscellaneous Propositions involving Unit Classes 
#54. Cardinal Couples .... 

#55. Ordinal Couples . . 
#56. The Ordinal Number 2 r . 

Section B. Sub-Classes, Sub-Relations, and Relative Types 

#60. The Sub-Classes of a given Class . 

#61. The Sub-Relations of a given Relation . 

#62. The Relation of Membership of a Class 

#63. Relative Types of Classes . . . 

#64. Relative Types of Relations 

#65. On the Typical Definition of Ambiguous Symbols 

Section C. One-Many, Many-One, and One-One Relations 

#70. Relations whose Classes of Referents and of Relata belong to 

given Classes . . . , ... 
#71. One-Many, Many-One, and One-One Relations . s . 
#72. Miscellaneous Propositions concerning One-Many, Many-One 

and One-One Relations . . . . 
#73. Similarity of Classes . 
#74. On One-Many and Many-One Relations with Limited Fields 



PAGE 

231 
232 
238 

242 
247 
256 
265 
277 
279 

296 
299 
302 
304 
315 
320 
324 

329 
331 
333 
340 
347 
352 
359 
366 
377 

386 
388 
393 
395 
400 
410 
415 

418 

420 
426 

441 
455 

468 



CONTENTS 



XI 



PAGE 

Section D. Selections ......... 478 

#80. Elementary Properties of Selections . . . . . 483 

#81. Selections from Many-One Relations ... . . . 496 

#82. Selections from Relative Products . . . . . 501 

#83. Selections from Classes of Classes ...... 508 

#84. Classes of Mutually Exclusive Classes . . . . . 517 

#85. Miscellaneous Propositions ....... 525 

#88. Conditions for the Existence of Selections .... 536 

Section E. Inductive Relations 543 

#90. On the Ancestral Relation ....... 549 

#91. On Powers of a Relation . . . . . . . 558 

#92. Powers of One-Many and Many-One Relations . . . 573 

#93. Inductive Analysis of the Field of a Relation . . . 579 

#94. On Powers of Relative Products ...... 588 

#95. On the Equi-factor Relation . . . . . . . 596 

#96. On the Posterity of a Term 607 

#97. Analysis of the Field of a Relation into Families . . . 623 

APPENDIX A 

#8. The Theory of Deduction for Propositions containing Apparent 

Variables . 635 

APPENDIX B 

#89. Mathematical Induction ....... 650 

APPENDIX C 

Truth-Functions and others . . . . . . . 659 

LIST OF DEFINITIONS 667 



ALPHABETICAL LIST OF PROPOSITIONS 
REFERRED TO BY NAMES 



\-~.py~p . D .~p 

b zq.D .j>vq 

bzp.pDq.^.q 

b zpv(qvr).3.qv(pvr) 

b z.p .D .gOr:3: q .3 ,p"Dr 

b z.pDq.p^r. D :p . D .q.r 

b .i.p.q.O.r : D :p. D . qDr 

f :. p "5q . D z p . r . D . q .r 

b.pDp 

b t.p . 3 . gO r : D zp~q . 3 . r 

i- : p v q . D . q v p 

b zq.D .pDq 

bzp.q.D.p 

b :p .q.^.q 

bz. qDr.Dzpvq.O.pvr 

b z.qDr.D zpDq.y.pDr 

b z.pOq.^'.q^r.D.pDr 

bz pDq.qOr.^.pDr 

b zqDr .pDq.'S.pDr 

b :pvp. D .p 

bzpD^q.D.q D~p 

b z <^>pDq . D ,<^q"Dp 

b z p D q . D . ~ q D ~ p 

b : ~ q D ~ p . D . p D q 

bz.p.q.D.rzDzp.^r."D.e^q 

b : p D q . = . ~ q "D ~ p 

bzp = q. = .<^p = <**>q 



Name 


Number 


Abs 


*201. 


Add 


*i*3. 


Ass 


*3 35. 


Assoc 


#1-5. 


Coram 


*2-04. 


Comp 


*3-43. 


Exp 


*33. 


Fact 


*3 45. 


Id 


*208. 


Imp 


*3'31. 


Perm 


#1-4. 


Simp 


*202. 


» 


*326. 


» 


*3*27. 


Sum 


*l-6. 


Syll 


*205. 


„ 


*206. 


» 


*333. 


» 


*3 34. 


Taut 


*l-2. 


Transp 


*203- 


» 


*215. 


5> 


*216. 


» 


*217. 


» 


*337. 


» 


*41. 


3J 


*411. 



INTRODUCTION TO THE SECOND EDITION* 

In preparing this new edition of Principia Mathematica, the authors have 
thought it best to leave the text unchanged, except as regards misprints and 
minor errors f, even where they were aware of possible improvements. The 
chief reason for this decision is that any alteration of the propositions would 
have entailed alteration of the references, which would have meant a very 
great labour. It seemed preferable, therefore, to state in an introduction the 
main improvements which appear desirable. Some of these are scarcely open 
to question ; others are, as yet, a matter of opinion. 

The most definite improvement resulting from work in mathematical logic 
during the past fourteen years is the substitution, in Part I, Section A, of the 
one indefinable "p and q are incompatible" (or, alternatively, "p and q are 
both false") for the two indefinables "not-p" and "p or q." This is due to 
Dr H. M. Sheffer]:. Consequentially, M. Jean Nicod§ showed that one 
primitive proposition could replace the five primitive propositions *1'2*3"4'5*6. 

From this there follows a great simplification in the building up of 
molecular propositions and matrices; #9 is replaced by a new chapter, #8, 
given in Appendix A to this Volume. 

Another point about which there can be no doubt is that there is no need 
of the distinction between real and apparent variables, nor of the primitive 
idea "assertion of a propositional function." On all occasions where, in 
Principia Mathematica, we have an asserted proposition of the form "V .fx" 
or "h .fp" this is to be taken as meaning "r- . (x) .fx " or " h . (p) .fp." Con- 
sequently the primitive proposition *1*11 is no longer required. All that is 
necessary, in order to adapt the propositions as printed to this change, is the 
convention that, when the scope of an apparent variable is the whole of the 
asserted proposition in which it occurs, this fact will not be explicitly indicated 
unless " some " is involved instead of " all." That is to say, "h . <f>x " is to mean 
" h . (x) . <fix " ; but in " I- . ( gar) . <f>x " it is still necessary to indicate explicitly 
the fact that " some " x (not " all " x's) is involved. 

It is possible to indicate more clearly than was done formerly what are 
the novelties introduced in Part I, Section B as compared with Section A. 

* In this introduction, as -well as in the Appendices, the authors are under great obligations 
to Mr F. P. Ramsey of King's College, Cambridge, who has read the whole in MS. and contributed 
valuable criticisms and suggestions. 

t In regard to these we are indebted to many readers, but especially to Drs Behmann and 
Boscovitch, of Gdttingen. 

X Tram. Amer. Math. Soc. Vol. xnr. pp. 481 — 488. 

§ "A reduction in the number of the primitive propositions of logic," Proc. Camb. Phil. Soc. 
Vol. MX. 



XIV INTRODUCTION 

They are three in number, two being essential logical novelties, and the third 
merely notational. 

(1) For the "p" of Section A, we substitute " <}>x," so that in place of 
" *- • (P) -fP " we have " h . (<}>, x) ./(<f>x)." Also, if we have " h ./{p, q, r, . . .)," 
we may substitute <f>x, <f>y, 4>z, ... for^, q, r, ... or <f>x, <f>y for p, q, and yfrz, ... 
for r, ..., and so on. We thus obtain a number of new general propositions 
different from those of Section A. 

(2) We introduce in Section B the new primitive idea " (g#) . <f>x," i.e. 
existence-propositions, which do not occur in Section A, In virtue of the 
abolition of the real variable, general propositions of the form " (p) . fp " do 
occur in Section A, but " (<&p) .fp " does not occur. 

(3) By means of definitions, we introduce in Section B general propositions 
which are molecular constituents of other propositions ; thus " (x) . <]>x . v . p " is 
to mean " (x) . <f>xvp." 

It is these three novelties which distinguish Section B from Section A. 

One point in regard to which improvement is obviously desirable is the 
axiom of reducibility (*12M1). This axiom has a purely pragmatic justifica- 
tion : it leads to the desired results, and to no. others. But clearly it is not 
the sort of axiom with which we can rest content. On this subject, however, 
it cannot be said that a satisfactory solution is as yet obtainable. Dr Leon 
Chwistek* took the heroic course of dispensing with the axiom without 
adopting any substitute ; from his work, it is clear that this course compels 
us to sacrifice a great deal of ordinary mathematics. There is another course, 
recommended by Wittgenstein f for philosophical reasons. This is to assume 
4hat functions of propositions are always truth-functions, and that a function 
can only occur in a proposition through its values. There are difficulties in 
the way of this view, but perhaps they are not insurmountable J. It involves 
the consequence that all functions of functions are extensional. It requires us 
to maintain that " A believes p " is not a function of p. How this is possible, 
is shown in Tractatus Logico-Philosophicus (loc. cit. and pp. 19—21). We are 
not prepared to assert that this theory is certainly right, but it has seemed 
worth while to work out its consequences in the following pages. It appears 
that everything in Vol. I remains true (though often new proofs are required) ; 
the theory of inductive cardinals and ordinals survives ; but it seems that the 
theory of infinite Dedekindian and well-ordered series largely collapses, so 
that irrationals, and real numbers generally, can no longer be adequately 
dealt with. Also Cantor's proof that 2 n > n breaks down unless n is finite. 
Perhaps some further axiom, less objectionable than the axiom of reducibility, 
might give these results, but we have not succeeded in finding such an axiom. 

* In his " Theory of Constructive Types." See references »t the end of this Introduction, 
j- Tractatus Logico-Philosophicus, *5*54 ff . 
X See Appendix C. 



INTRODUCTION XV 

It should be stated that a new and very powerful method in mathematical 
logic has been invented by Dr H. M. Sheflfer. This method, however, would 
demand a complete re-writing of Principia Mathematica. We recommend 
this task to Dr Sheffer, since what has so far been published by him is 
scarcely sufficient to enable others to undertake the necessary reconstruction. 

We now proceed to the detailed development of the above general sketch. 

I. ATOMIC AND MOLECULAR PROPOSITIONS 

Our system begins with "atomic propositions." We accept these as a 
datum, because the problems which arise concerning them belong to the 
philosophical part of logic, and are not amenable (at any rate at present) to 
mathematical treatment. 

Atomic propositions may be defined negatively as propositions containing 
no parts that are propositions, and not containing the notions "all" or "some." 
Thus " this is red," "this is earlier than that," are atomic propositions. 

Atomic propositions may also be defined positively — and this is the better 
course — as propositions of the following sorts r 

R 1 (x), meaning "x has the predicate R^'; 

R*( x >y) [° r xRzy]' meaning "x has the relation R 2 (in intension) to y"; 

R 3 (x,y, z), meaning "x,y,z have the triadic relation R 3 (in intension)"; 

R 4 (x, y, z, w), meaning "x,y,z,w have the tetradic relation R 4 (in intension)"; 
and so on ad infinitum, or at any rate as 'long as possible. Logic does not 
know whether there are in fact n-adic relations (in intension); this is an empirical 
question. We know as an empirical fact that there are at least dyadic relations 
(in intension), because without them series would be impossible. But logic is 
not interested in this fact; it is concerned solely with the hypothesis of there 
being propositions of such-and-such a form. In certain cases, this hypothesis is 
itself of the form in question, or contains a part which is of the form in question ; 
in these cases, the fact that the hypothesis can be framed proves that it is 
true. But even when a hypothesis occurs in logic, the fact that it can be 
framed does not itself belong to logic. 

Given all true atomic propositions, together with the fact that they are all, 
every other true proposition can theoretically be deduced by logical methods. 
That is to- say, the apparatus of crude fact required in proofs can all be con- 
densed into the true atomic propositions together with the fact that every 
true atomic proposition is one of the following: (here the list should follow). 
If used, this method would presumably involve an infinite enumeration, 
since it seems natural to suppose that the number of true atomic propositions 
is infinite, though this should not be regarded as certain. In practice, 
generality is not obtained by the method of complete enumeration, because 
this method requires more knowledge than we possess. 

R&W I h 



Xvi INTRODUCTION 

We must now advance to molecular propositions. Let p, q, r, s, t denote, 
to begin with, atomic propositions. We introduce the primitive idea 

which may be read "p is incompatible with q"* and is to be true whenever 
either or both are false. Thus it may also be read "p is false or q is false"; 
or again, "p implies not-q." But as we are going to define disjunction, impli- 
cation, and negation in terms of p | q, these ways of reading p \ q are better 
avoided to begin with. The symbol "p\ q" is pronounced: "p stroke q." We 
now put 



~P . = -P\P 


Df, 


pD q . = -p|~<7 


Df, 


pv q . = .^pl^q 


Df, 


p.q. = .~(p\q) 


Df. 



Thus all the usual truth-functions can be constructed by means of the stroke. 

Note that by the above, 

p3q. = .p\(q\q) Df. 
We find that 

p.D.q.r. = .p\(q\r). 

Thus p D q is a degenerate case of a function of three propositions. 

We can construct new propositions indefinitely by means of the stroke ; 
for example, (p \ q) j r, p \ (q \ r), (p | q) | (r\s), and so on. Note that the stroke obeys 
the permutative law (p \ q) = (q \p) but not the associative law (p\q)\r =p\(q\r). 
(These of course are results to be proved later.) Note also that, when we 
construct a new proposition by means of the stroke, we cannot know its truth 
or falsehood unless either (a) we know the truth or falsehood of some of its 
constituents, or (b) at least one of its constituents occurs several times in a 
suitable manner. The case (a) interests logic as giving rise to the rule of in- 
ference, viz. 

Given p and p \ (q \ r), we can infer r. 

This or some variant must be taken as a primitive proposition. For the 
moment, we are applying it only when p, q, r are atomic propositions, but we 
shall extend it later. We shall consider (6) in a moment. 

In constructing new propositions by means of the stroke, we assume that 
the stroke can have on either side of it any proposition so constructed, and 
need not have an atomic proposition on either side. Thus given three atomic 
propositions p, q, r, we can form, first, p \ q and q \ r, and thence (p\q)\ r and 
p | (q | r). Given four, p, q, r, s, N we can form 

{(p\q)\r}\s, (p\q)\(r\s), p\{q\(r\s)} 
and of course others by permuting p, q, r, s. The above three are substantially 

* For what follows, see Nicod, " A reduction in the number of the primitive propositions of 
logic," Proc. Camb. Phil. Soc. Vol. xix. pp. 32—41. 



INTRODUCTION XV11 

different propositions. We have in fact 

{(P I <l) H I s • - '-^P v ~<7 • r : v :~s, 
(p\q)\(r\s). = :p.q.v.r.s, 
p\{q\(r\s)} . = :.~p : V : q .~rv~*. 
All the propositions obtained by this method follow from one rule: in 
"P I q" substitute, for p or q or both, propositions already constructed by means 
of the stroke. This rule generates a definite assemblage of new propositions 
out of the original assemblage of atomic propositions. All the propositions so 
generated (excluding the original atomic propositions) will be called " mole- 
cular propositions." Thus molecular propositions are all of the form p \ q, but 
the p and q may now themselves be molecular propositions. If p is p 1 \p 2 , 
p x and p 2 may be molecular; suppose Pi = pn\pi2- Pn may be of the form 
Piu \Piu, an d so on; but after a finite number of steps of this kind, we are to 
arrive at atomic constituents. In a proposition/) | q, the stroke between p and 
q is called the "principal" stroke; if p=p x \p 2 , the stroke between p x and p 2 is 
a secondary stroke; so is the stroke between q x and q 2 if q = q x \ q 2 . If pi =p u | p u , 
the stroke between p n and p 12 is a tertiary stroke, and so on. 

Atomic and molecular propositions together are " elementary propositions." 
Thus elementary propositions are atomic propositions together with all that 
can be generated from them by means of the stroke applied any finite number 
of times. This is a definite assemblage of propositions. We shall now, until 
further notice, use the letters p, q, r, s, t to denote elementary propositions, 
not necessarily atomic propositions. The rule of inference stated above is to 
hold still; i.e. 

If p, q, r are elementary propositions, given p and p | (q | r), we can infer r. 

This is a primitive proposition. 

We can now take up the point (6) mentioned above. When a molecular 
proposition contains repetitions of a constituent proposition in a suitable 
manner, it can be known to be true without our having to know the truth or 
falsehood of any constituent. The simplest instance is 

P\(P\P)> 
which is always true. It means "p is incompatible with the incompatibility 
of p with itself," which is obvious. Again, take "p . q . D . p." This is 

{(p \q)\(p\ q)\ I (P I P)- 

Again, take "~jp.D.~pv~ q." This is 

(p\p)\ {(p \q)\(p\ q)}> 

Again, "p . D .p v q " is 

p\i{(p\p)\(q\q)}\i(p\p)\(q\q)}l 

All these are true however p and q may be chosen. It is the fact that we can 
build up invariable truths of this sort that makes molecular propositions 
important to logic. Logic is helpless with atomic propositions, because their 

62 



XV111 INTRODUCTION 

truth or falsehood can only be known empirically. But the truth of molecular 
propositions of suitable form can be known universally without empirical 
evidence. 

The laws of logic, so far as elementary propositions are concerned, are all 
assertions to the effect that, whatever elementary propositions p, q, r, ... may 
be, a certain function 

F(p,q,r,...), 

whose values are molecular propositions, built up by means of the stroke, is 
always true. The proposition " F(p) is true, whatever elementary proposition 
p may be " is denoted by 

(p).F(p). 

Similarly the proposition "F(p,q,r,...) is true, whatever elementary pro- 
positions p, q, r, ... may be " is denoted by 

(p,q,r, ...).F(p,q,r, ...). 

When such a proposition is asserted, we shall omit the "(p,q,r, ...)" at the 
beginning. Thus 

"\-.F{p,q,r,...V 

denotes the assertion (as opposed to the hypothesis) that F(p,q,r, ...) is true 
whatever elementary propositions p, q, r, ... may be. 

(The distinction between real and apparent variables, which occurs in 
Frege and in Principia Mathematica, is unnecessary. "Whatever appears as a 
real variable in Principia Mathematica is to be taken as an apparent variable 
whose scope is the whole of the asserted proposition in which it occurs.) 

The rule of inference, in the form given above, is never required within 
logic, but only when logic is applied. Within logic, the rule required is different. 
In the logic of propositions, which is what concerns us at present, the rule 
used is : 

Given, whatever elementary propositions p, q, r may be, both 
"K F(p, q, r,. ..)" and "h .F(p,q,r, ...)\{G(p,q,r, ...)\H(p, q,r, . ..)}," 
we can infer " h . H{p, q, r, ...)." 

Other forms of the rule of inference will meet us later. For the present, 
the above is the form we shall use. 

Nicod has shown that the logic of propositions (*1 — *5) can be deduced, 
by the help of the rule of inference, from two primitive propositions 

and \- m .pDq.D.s\qDp\s. 

The first of these may be interpreted as "p is incompatible with not-p," or 
as "p or not-p," or as "not (p and not-p)," or as "p implies p." The second 
may be interpreted as 

pDq.DzqDf^s.D.pD^s, 



INTRODUCTION XIX 

which is a form of the principle of the syllogism. Written wholly in terms of 
the stroke, the principle becomes 

{p\(9\q)}\[{(s\q)\((p\s)\(p\s))}\{(s\q)\((p\s)\(p\s))}]. 

Nicod has shown further that these two principles may be replaced by 
one. Written wholly in terms of the stroke, this one principle is 

bl(g|r)}|[{*|(*|*)}|K«l«)l((pl*)l(pl«))}]- 

It will be seen that, written in this form, the principle is less complex than 

the second of the above principles written wholly in terms of the stroke. 

When interpreted into the language of implication, Nicod's one principle 

becomes 

p.0.q.r:^.tDt.s\qDp\s. 

In this form, it looks more complex than 

pDj.D .s\qDp\s, 

but in itself it is less complex. 

From the above primitive proposition, together with the rule of inference, 
everything that logic can ascertain about elementary propositions can be 
proved, provided we add one other primitive proposition, viz. that, given a 
proposition (p, q, r, ...) . F (p, q> r, ...), we may substitute for p, q, r, ... 
functions of the form 

/,0>, ?, r, ...), f \ (p, q,r ,...), f s (p, q, r, ...) 
and assert 

(p,q,r,...).FUi(p,q,r, ...), f 3 (p,q,r, ...),f 3 (p,q,r, ...), ...}, 
where f 1} / 2 , f 3 , ... are functions constructed by means of the stroke. Since 
the former assertion applied to all elementary propositions, while the latter 
applies only to some, it is obvious that the former implies the latter. 

A more general form of this principle will concern us later. 

II. ELEMENTARY FUNCTIONS OF INDIVIDUALS 
1. Definition of '" individual" 

We saw that atomic propositions are of one of the series of forms: 
R x {x), R^{x,y), R 3 (x,y,z), R^{x,y,z y w\ .... 
Here R lt R 2 , R s , R 4 , ... are each characteristic of the special form in which 
they are found: that is to say, R n cannot occur in an atomic proposition 
R m (x 1} # 2 > ••• #m) unless n = m, and then can only occur as R m occurs, not as 
x lt x 2 , ... x m occur. On the other hand, any term which can occur as the 
a-'s occur in R n (x 1} x 2 , ... x n ) can also occur as one of the x's in R m (x x , x 2 , . . . x m ) 
even if m is not equal to n. Terms which can occur in any form of atomic 
proposition are called " individuals" or " particulars"; terms which occur as the 
R's occur are called " universals." 

We might state our definition compendiously as follows: An " individual" 
is anything that can be the subject of an atomic proposition. 



XX INTRODUCTION 

Given an atomic proposition R n (x 1} x 2 , ... x n ), we shall call any of the x's 
a "constituent" of the proposition, and R n a " component " of the proposition*. 
We shall say the same as regards any molecular proposition in which 
R n (x 1} x 2 , ... x n ) occurs. Given an elementary proposition p j q, where p and q 
may be atomic or molecular, we shall call p and q " parts " of p | q; and any 
parts of p or q will in turn be called parts of p | q, and so on until we reach the 
atomic parts of p \ q. Thus to say that a proposition r " occurs in" p \ q and to 
say that r is a "part " of p | q will be synonymous. 

2. Definition of an elementary function of an individual 

Given any elementary proposition which contains a part of which an 
individual a is a constituent, other propositions can be obtained by replacing 
a by other individuals in succession. We thus obtain a certain assemblage 
of elementary propositions. We may call the original proposition 0a, and 
then the propositional function obtained by putting a variable x in the 
place of a will be called <f>x. Thus <f>x is a function of which the argument 
is x and the values are elementary propositions. The essential use of "<f>x" 
is that it collects together a certain set of propositions, namely all those that 
are its values with different arguments. 

We have already had various special functions of propositions. If p is a 
part of some molecular proposition, we may consider the set of propositions 
resulting from the substitution of other propositions for p. If we call the 
original molecular proposition fp, the result of substituting q is called /#. 

When an individual or a proposition occurs twice in a proposition, three 
functions can be obtained, by varying only one, or only another, or both, of 
the occurrences. For example, p \p is a value of any one of the three functions 
P I <?> 9 1 P> 9 I 9> where q is the argument. Similar considerations apply when an 
argument occurs more than twice. Thus p\(p\p) is a value of q\(r\s), or 
9 ! ( r I <l)> or 9 i (<? 1 r), or q\(r\ r), or q\(q\ q). When we assert a proposition 
" ^ • (P) » Fp," the p is to be varied whenever it occurs. We may similarly 
assert a proposition of the form " (x) . <f>x," meaning " all propositions of the 
assemblage indicated by <f>x are true"; here also, every occurrence of x is to be 
varied. 

• 3. "Always true" and "sometimes true" 

Given any function, it may happen that all its values are true; again, it 
may happen that at least one of its values is true. The proposition that all 
the values of a function (x,y, z, ...) are true is expressed by the symbol 

"(x,y,z, ...).$ (x,y,z,...)" 
unless we wish to assert it, in which case the assertion is written 

"h.(f)(x,y,z, ...)." 
* This terminology is taken from Wittgenstein. 



INTRODUCTION XXI 

We have already had assertions of this kind where the variables were ele- 
mentary propositions. We want now to consider the case where the variables 
are individuals and the function is elementary, i.e. all its values are elementary 
propositions. We no longer wish to confine ourselves to the case in which it 
is asserted that all the values of <f>(x,y,z, ...) are true; we desire to be able 
to make the proposition 

(x } y,z,...).<\>{x,y,z, ...) 
a part of a stroke function. For the present, however, we will ignore this 
desideratum, which will occupy us in Section III of this Introduction. 

In addition to the proposition that a function $x is "always true" 
(i.e. (x) . <j>x), we need also the proposition that <f>x is " sometimes true," i.e. is 
true for at least one value of x. This we denote by 

"(a*) • <K' 

Similarly the proposition that <f> (x, y, z, . . .) is "sometimes true" is denoted by 

il (^x,y,z, ...).4>(x,y,z,...)." 

We need, in addition to (x, y, z, . . . ) . <f> (x, y,z,...) and (3a;, y, z, ...).<£(#, y, z, ... ), 
various other propositions of an analogous kind. Consider first a function of 
two variables. We can form 

(a*) : (y) • <i> fo y)> O) : (as/) ■ <t> ( x > y)> (as/) = (#)■<£ («, y)> (y) • (a*) ■ <t> (*» y)- 

These are substantially different propositions, of which no two are always 
equivalent. It would seem natural, in forming these propositions, to regard 
the function £ (x, y) as formed in two stages. Given <f> (a, b), where a and b 
are constants, we can first form a function <f> (a, y), containing the one variable 
y; we can then form 

(y ) . <f> ( a, y) and (33/) . </> (a , y). 
We can now vary a, obtaining again a function of one variable, and leading 
to the four propositions 

(x) :(y).<f> (x, y), ( a a>) : (y) . <f> {x, y), (x) : (ay) . <f> (x, y), (gar) : (33/) • <\> 0*. 2/)- 
On the other hand, we might have gone from </> (a, b) to <f> (x, b), thence to 
(x) . <j> (x, b) and (3a;) . <f> (x, 6), and thence to 

(y) : {as) . <j> (x, y), ( 3 y) :(*).£ (x, y), (y) : (a*) . <f> (x, y), (ay) : (3*0 • 0*. 2/)- 

All of these will be called "general propositions"; thus eight general 
propositions can be derived from the function <f> (x, y). We have 
(x) : (y) . <f> (x, y) : = : (y) : («) . 4> (a?, y), 

(a«) : (ay) • <f> te 2/) : = : (ay) = (a*0 ■ 4> te y)- 

But there are no other equivalences that always hold. For example, the dis- 
tinction between " (x) : (gy) . <j> (x, y) " and " (gy) : (x) . <f> (x, y) " is the same 
as the distinction in analysis between " For every e, however small, there is a 
8 such that..." and " There is a 8 such that, for every e, however small " 



XX11 INTRODUCTION 

Although it might seem easier, in view of the above considerations, to 
regard every function of several variables as obtained by successive steps, each 
involving only a function of one variable, yet there are powerful considerations 
on the other side. There are two grounds in favour of the step-by-step method ; 
first, that only functions of one variable need be taken as a primitive idea; 
secondly, that such definitions as the above seem to require either that we 
should first vary x, keeping y coostant, or that we should first vary y, keeping 
x constant. The former seems to be involved when " (y) " or " fay) " appears 
to the left of " (x) " or " fax)," the latter in the converse case. The grounds 
against the step-by-step method are that it interferes with the method of 
matrices, which brings order into the successive generation of types of pro- 
positions and functions demanded by the theory of types, and that it requires 
us, from the start, to deal with such propositions as (y) . <f> (a, y), which are 
not elementary. Take, for example, the proposition " h : q . D . p v q." This 
will be 

\-:.(p):.(q):q.D .pvq, 
or h:.(q):.(p):q.D.pvq, 

and will thus involve all values of either 

(q) : q . D . p v q considered as a function of p, 
or (p) :q.D .pvq considered as a function of q. 

This makes it impossible to start our logic with elementary propositions, as 
we wish to do. It is useless to enlarge the definition of elementary propositions, 
since that only increases the values of q or p in the above functions. Hence 
it seems necessary to start with an elementary function 

(pi&l, x%, x 3 , ... x n ), 
before which we write, for each x r , either "(x r )" or " fax r )," the variables in 
this process being taken in any order we like. Here <f> {x ly x 2i x 3 , ... x n ) is 
called the " matrix," and what comes before it is called the " prefix." Thus in 

(a^) : (y) • 4> 0> y) 

" <f> (x, y) " is the matrix and " fax) : (y) " is the prefix. It thus appears that 
a matrix containing n variables gives rise to n 1 2 n propositions by taking its 
variables in all possible orders and distinguishing " (x r ) " and " fax r ) " in each 
case. (Some of these, however, are equivalent.) The process of obtaining such 
propositions from a matrix will be called " generalization," whether we take 
" all values " or " some value," and the propositions which result will be called 
" general propositions." 

We shall later have occasion to consider matrices containing variables that 
are not individuals ; we may therefore say : 

A " matrix " is a function of any number of variables (which may or may 
not be individuals), which has elementary propositions as its values, and is 
used for the purpose of generalization. 



INTRODUCTION XX111 

A " general proposition " is one derived from a matrix by generalization. 
We shall add one further definition at this stage : 

A " first-order proposition " is one derived by generalization from a matrix 
in which all the variables are individuals. 

4. Methods of proving general propositions 

There are two fundamental methods of proving general propositions, one 
for universal propositions, the other for such as assert existence. The method 
of proving universal propositions is as follows. Given a proposition 

«\-.F(p,q,r,...y 
where F is built up by the stroke, and p, q,r, ... are elementary, we may re- 
place them by elementary functions of individuals in any way we like, putting 

P == Ji\p l h) &?> '■'• ®n)> 
q z =j2\ x ii x 2> ••• x n)> 

and so on, and then assert the result for all values of x lt x 2 , ... x n . What we 
thus assert is less than the original assertion, since p, q, r, ... could originally 
take all values that are elementary propositions, whereas now they can only 
take such as are values of /i,/ 2 ,/ 3 , — (Any two or more of /i,/ 2 ,/ 3 , ... may 
be identical.) 

For proving existence- theorems we have two primitive propositions, namely 
#81. I- . (g#, y) . <f>a | (<f>a; | <f>y) and 

#811. I- . fax) . <f>x | (<pa | <f>b) 

Applying the definitions to be given shortly, these assert respectively 

<pa . D . fax) . <px 
and (x) . <j>x . D . <j>a . <f>b. 

These two primitive propositions are to be assumed, not only for one variable, 
but for any number. Thus we assume 

<f> (<*!, a 2 , ... a n ) . D . (g#i, x 2 , ... x n ) . <f> {x 1> x 2 , ... x n ), 
(x 1} x 2 , ... x n ). <£(#i, # 2 , ... x n ). D. ^>(ai, Oa, ... a»). <£(&i, & 2 > ••■ &»)• 
The proposition (x) . <f>x . D . <f>a . <f>b, in this form, does not look suitable for 
proving existence-theorems. But it may be written 

(g#) . ~ <f>x . v . <f>a . <f>b 
or ~ <j>a v ~ <f)b . D . fax) . ~ <f>x, 

in which form it is identical with #911, writing <f> for ~^>. Thus our two 
primitive propositions are the same as #91 and #911. 

For purposes of inference, we still assume that from (x) . <j>x and 
(x) . <f>x D yfrx we can infer (x) . yfrx, and from p and p D q we can infer q, even 
when the functions or propositions involved are not elementary. 



XXIV INTRODUCTION 

Existence-theorems are very often obtained from the above primitive 
propositions in the following manner. Suppose we know a proposition 

\-.f(x,x). 
Since <f>x . D . fay) . <f>y, we can infer 

May) -/toy). 

i.e. H:(#):(a2/)./(a;,y). 

Similarly r : (y) : fax) .f(x, y). 

Again, since <j> (x, y) . D • faz, w) . <£> (z, w\ we can infer 

•■ ■ (a^ y) ■"/(*» y) 

and ' H. (ay, «)•/(*, y). 

We may illustrate the proofs both of universal and of existence propo- 
sitions by a simple example. We have 

Hence, substituting <f>x for p, 

h . {x) . <f>x D <f)X. 

Hence, as in the case of /(#, x) above, 

t- rfa) : (ay) ■ fa D <f>y, 

b : (y) : fax) . <f>x D cf>y, 

I" ■ fax, y)-fa^ 4>y- 

Apart from special axioms asserting existence-theorems (such as the axiom of 
reducibility, the multiplicative axiom, and the axiom of infinity), the above 
two primitive propositions give the sole method of proving existence-theorems 
in logic. They are, in fact, always derived from general propositions of the 
form (x).f(x,x) or (x) ,f(x,x,x) or etc., by substituting other variables for 
some of the occurrences of x. 

III. GENERAL PROPOSITIONS OF LIMITED SCOPE 

In virtue of a primitive proposition, given (x) . <f>x and (x) . $x D -tyx, we 
can infer (x) . yjrx. So far, however, we have introduced no notation which 
would enable us to state the corresponding implication (as opposed to inference). 
Again, fax) . §x and (x, y) . $x O yfry enable us to infer (y) . tyy; here again, 
we wish to be able to state the corresponding implication. So far, we have only 
defined occurrences of general propositions as complete asserted propositions. 
Theoretically, this is their only use, and there is no need to define any other. 
But practically, it is highly convenient to be able to treat them as parts 
of stroke-functions. This is entirely a matter of definition. By introducing 
suitable definitions, first-order propositions can be shown to satisfy all the 
propositions of #1 — *5. Hence in using the propositions of #1— #5, it will 
no longer be necessary to assume that p, q, r, ... are elementary. 

The fundamental definitions are given below. 



INTRODUCTION XXV 

When a general proposition occurs as part of another, it is said to have 
limited scope. If it contains an apparent variable x, the scope of x is said to 
be limited to the general proposition in question. Thus in p \ {(x) . <f>x\, the 
scope of x is limited to ix) . <ftx, whereas in (x) . p | fa the scope of x extends 
to the whole proposition. Scope is indicated by dots. 

The new chapter *8 (given in Appendix A) should replace *9 in Principia 
Mathematica. Its general procedure will, however, be explained now. 

The occurrence of a general proposition as part of a stroke-function is 
denned by means of the following definitions: 

{(x).<j>x}\q. = .fax).<f>x\q Df, 

1(3*0 • fa) 1 9. ■ = ■ (*) ■ fa 1 ? Df > 

p I {(ay) • tyy) • = ■ (y) • v \ fy Df - 

These define, in the first place, only what is meant by the stroke when it 
occurs between two propositions of which one is elementary while the other is 
of the first order. When the stroke occurs between two propositions which 
are both of the first order, we shall adopt the convention that the one on the 
left is to be eliminated first, treating the one on the right as if it were ele- 
mentary; then the one on the right is to be eliminated, in each case, in 
accordance with the above definitions. Thus 

{{x) . <f>x} | [(y) . yjry] . = : fax) : <f>x\ {(y) . ^y} : 
= = (3*0 = (32/) ■ fa I t2A 

(0) • fa] I Kay) •■fy}- = - (3*0 = fa I {(ay) ■ -M = 
= = (3*0 : (y) • fa I tyy> 

{fax) . <f>x} | \{y) .^ry}.= : (x) : fay) . <f>x j fy. 

The rule about the order of elimination is only required for the sake of 
definiteness, since the two orders give equivalent results. For example, in 
the last of the above instances, if we had eliminated y first we should have 
obtained 

(ay) : (*0 ■ fa I ^y> 

which requires either (x) ,<^>$x or fay) .<^-tyy, and is then true. 

And (x) : fay) . <f>x | yfry 

is true in the same circumstances. This possibility of changing the order of 
the variables in the prefix is only due to the way in which they occur, i.e. to 
the fact that x only occurs on one side of the stroke and y only on the other. 
The order of the variables in the prefix is indifferent whenever the occurrences 
of one variable are all on one side of a certain stroke, while those of the other 
are all on the other side of it. We do not have in general 

(a*0 : (y) • x ( x > y)- = -iy)' (3*0 ■ x 0»> y); 



XXVI INTRODUCTION 

here the right-hand side is more often true than the left-hand side. But we 
do have 

(ft®) '• iy) -<l>a!\yjry: = : (y) : (a*) . $x\^y. 
The possibility of altering the order of the variables in the prefix when they 
are separated by a stroke is a primitive proposition. In general it is convenient 
to put on the left the variables of which "all" are involved, and on the right 
those of which " some " are involved, after the elimination has been finished, 
always assuming that the variables occur in a way to which our primitive 
proposition is applicable. 

It is not necessary for the above primitive proposition that the stroke 
separating x and y should be the principal stroke, e.g. 

p I [{(a*) • <H I {(y) • "f y}] ■ = • p I [0*0 : (ay) • 4> x I ^y] ■ 

(a*) : (y) • p I (<£* I iry) - 
(y) • (a«) • p I {<t> x 1 1ry)> 

All that is necessary is that there should be some stroke which separates x 

from y. When this is not the case, the order cannot in general be changed. 

Take e.g. the matrix 

<f>x V yjry . ~ <f>x V <^» ifry. 

This may be written (<j>x D yjry) j {$-y D <f>x) 

or {fx | (fy | tyy)} \ [tyy \ (Qx | <f>x)}. 

Here there is no stroke which separates all the occurrences of x from all those 
of y, and in fact the two propositions 

(y) ' (a 57 ) • § x v "tyy ■ ~ ^ v ^ ^y 

and (a«) : (y) . 4>x v tyy .~^pv<v yfry 

are not equivalent except for special values of <f> and i|r. 

By means of the above definitions, we are able to derive all propositions, 
of whatever order, from a matrix of elementary propositions combined by 
means of the stroke. Given any such matrix, containing a part p, we may 
replace p by <f>x or <f> (x, y) or etc., and proceed to add the prefix (x) or (g#) 
or (x, y) or (x) : (gy) or (y) : (gp) or etc. If p and q both occur, we may replace 
p by <f)X and q by tyy, or we may replace both by <j>%, or one by <f>x and another 
by some stroke-function of <f>x. 

In the case of a proposition such as 

p I (O) = (ay) ■ * (®> y) ] > 

we must treat it as a case of p \ {(x) . </>#}, and first eliminate x. Thus 

p I {(«) : (ay) ■ f («» y)} • = : (a*) -(y)-p\ ^0*»y)« 

That is to say, the definitions of {(x) . <f>x)} \ q etc. are to be applicable un- 
changed when <f>x is not an elementary function. 



INTRODUCTION 



xxvu 



The definitions of ~ 
Thus 

•~ {(x) . <f>x\ . = 



p, pv q, p .q, pOq are to be taken over unchanged. 



p . D . (x) .</>#: = 



(x) . <f>x . D . p : = 
(x) . <f>x . v . p : = 

p . v . (x) . <f>x : = 



{(x) . fa] I \(x) . <f>x} : 
(rx) : <f>x | {(x): <f>x\ : 

(a«0 • (33/) • (<£# 1 4>y\ 

(x) : (y) . fox | <£y), 

p'l [{<*)■**} I {(*)■**}]: 
P I {(3*0 = (33/) • (4& I #)} = 
(x) : (y) .p | (<f>x | <£y), 

{(#) . <f>x] \(p\p): 

(rx) . <f>x | (p | p) : = : (gar) .<f>xDp, 

[~{(»).^}]| ~p: 

Ka«) : (ay) • (^ I <&/)} i (/> I P) : 
(^)-{(ay)-(^l^)}|(i>lp): 

(*):(y).(<M&/)KH.P)> 
(x):(y).(p\p)\(<f>x\<f>y). 

It will be seen that in the above two variables appear where only one might 
have been expected. We shall find, before long, that the two variables can be' 
reduced to one ; i.e. we shall have 

(3*0 : (32/) - <£# I <f>V '• = • (3«) • 4> x I £*» 
(a;) : (y) . <f>x \ <f>y : = . (a;) . <£# | <j>x. 
These lead to 

~ {(x) . <j)x} . = . (a«) . ~ fa, 
~ {(a^) • 4*®} ■ = ■(#)■ ~ <j>x. 

But we cannot prove these propositions at our present stage ; nor, if we could, 
would they be of much use to us, since we do not yet know that, when two 
general propositions are equivalent, either may be substituted for the other 
as part of a stroke-proposition without changing the truth-value. 

For the present, therefore, suppose we have a stroke-function in which p 
occurs several times, say p | (p | p), and we wish to replace p by (x) . <f>x, we 
shall have to write the second occurrence of p " (y) . <f>y," and the third 
" (z) . <f>z." Thus the resulting proposition will contain as many separate 
variables as there are occurrences of p. 

The primitive propositions required, which have been already mentioned, 
are four in number. They are as follows: 

(1) I- . (a», y) . $a | (<f>x | <j>y), i.e. \-:<f>a.D . (ga?) . <f>x. 

(2) I- . (g#) . <f>x | (<fxi j <f>b), i.e. H : (x) . <f>x . D . <f>a . <f>b. 

(3) The extended rule of inference, i.e. from (x) . <f>x and (x) . <j>x D i/r# 
we can infer (x) . tyx, even when <£ and yfr are not elementary. 

(4) If all the occurrences of x are separated from all the occurrences of 
y by a certain stroke, the order of x and y can be changed in the prefix; i.e. 



XXViii INTRODUCTION 

For (g#) : (y) . <f>x \ -fy we can substitute (y) : (ga;) . <f>% | yjry, and vice 
versa, even when this is only a part of the whole asserted proposition. 

The above primitive propositions are to be assumed, not only for one 
variable, but for any number. 

By means of the above primitive propositions it can be proved that all 

the propositions of #1 — *5 apply equally when one or more of the propositions 

p,q,r t ... involved are not elementary. For this purpose, we make use of the 

work of Nicod, who proved that the primitive propositions of *I can all be 

deduced from 

h .p Op 

and b .pDq.D .s\qDp\s 

together with the rule of inference: " Given p and p\(q\ r), we can infer r." 

Thus all we have to do is to show that the above propositions remain true 

when p, q, s, or some of them, are not elementary. This is done in #8 in 

Appendix A r 

IV. FUNCTIONS AS VARIABLES 

The essential use of a variable is to pick out a certain assemblage of 
elementary propositions, and enable us to assert that all members of this 
assemblage are true, or that at least one member is true. We have already 
used functions of individuals, by substituting <j>x for p in the propositions of 

#1 #5 7 and by the primitive propositions of #8. But hitherto we have always 

supposed that the function is kept constant while the individual is varied, and 
we have not considered cases where we have "g</>," or where the scope of "<]>" 
is less than the whole asserted proposition. It is necessary now to consider 
such cases. 

Suppose a is a constant. Then "<j>a" will denote, for the various values 
of <f>, all the various elementary propositions of which a is a constituent. This 
is a different assemblage of elementary propositions from any that can be 
obtained by variation of individuals; consequently it gives rise to new general 
propositions. The values of the function are still elementary propositions, 
just as when the argument is an individual; but they are a new assemblage 
of elementary propositions, different from previous assemblages. 

As we shall have occasion later to consider functions whose values are not 
elementary propositions, we will distinguish those that have elementary 
propositions for their values by a note of exclamation between the letter 
denoting the function and the letter denoting the argument. Thus "<£ ! x" is 
a function of two variables, x and </> ! £. It is a matrix, since it contains no 
apparent variable and has elementary propositions for its values. We shall 
henceforth write "<£ ! x" where we have hitherto written <j>x. 

If we replace a? by a constant a, we can form such propositions as 
(<f>).cf>l a, (a<£) . <f> ! a. 



INTRODUCTION XXIX 

These are not elementary propositions, and are therefore not of the form </> ! a. 
The assertion of such propositions is derived from matrices by the method of 
#8. The primitive propositions of #8 are to apply when the variables, or some 
of them, are elementary functions as well as when they are all individuals. 

A function can only appear in a matrix through its values*. To obtain a 
matrix, proceed, as before, by writing <f> ! x, i/r ! y, % I z, . .. in place of p, q, r, ... 
in some molecular proposition built up by means of the stroke. We can then 

apply the rules of *8 to <f>, ty, %, . . . as well as to x, y, z, The difference 

between a function of an individual and a function of an elementary function 
of individuals is that, in the former, the passage from one value to another 
is effected by making the same statement about a different individual, while 
in the latter it is effected by making a different statement about the same 
individual. Thus the passage from "Socrates is mortal" to "Plato is mortal" 
is a passage from/! x to fly, but the passage from "Socrates is mortal" to 
"Socrates is wise" is a passage from <j> I a to yjr ! a. Functional variation is 
involved in such a proposition as: "Napoleon had all the characteristics of a 
great general." 

Taking the collection of elementary propositions, every matrix has values 
all of which belong to this collection. Every general proposition results from 
some matrix by generalization f. Every matrix intrinsically determines a 
certain classification of elementary propositions, which in turn determines the 
scope of the generalization of that matrix. Thus " x loves Socrates " picks out 
a certain collection of propositions, generalized in " (x) . x loves Socrates " and 
"(qx) . x loves Socrates." But " <f> ! Socrates" picks out those, among elementary 
propositions, which mention Socrates. The generalizations "(<£) . <f> ! Socrates" 
and " (a0) . </> ! Socrates " involve a class of elementary propositions which 
cannot be obtained from an individual- variable. But any value of "<j> ! Socrates " 
is an ordinary elementary proposition ; the novelty introduced by the variable 
^> is a novelty of classification, not of material classified. On the other hand, 
(x) . x loves Socrates, (<£) . (f> ! Socrates, etc. are new propositions, not contained 
among elementary propositions. It is the business of #8 to show that these 
propositions obey the same rules as elementary propositions. The method of 
proof makes it irrelevant what the variables are, so long as all the functions 
concerned have values which are elementary propositions. The variables may 
themselves be elementary propositions, as they are in #1 — #5. 

A variable function which has values that are not elementary propositions 
starts a new set. But variables of this sort seem unnecessary. Every elementary 
proposition is a value of </> ! & ; therefore 

(p) .fp. = . (<£, *)./(* ! x) : (gp) . fp . = . (a</>, x) ./(<£ ! x). 

* This assumption is fundamental in the following theory. It has its difficulties, but for the 
moment we ignore them. It takes the place (not quite adequately) of the axiom of reducibility. 
It is discussed in Appendix C. 

f In a proposition of logic, all the variables in the matrix must be generalized. In other 
general propositions, such as "all men are mortal," some of the variables in the matrix are re- 
placed by constants. 



XXX INTRODUCTION 

Hence all second-order propositions in which the variable is an elementary 
proposition can be derived from elementary matrices. The question of other 
second-order propositions will be dealt with in the next section. A function 
of two variables, say <f> (x, y), picks out a certain class of classes of propositions. 
We shall have the class <f> (a, y), for given a and variable y ; then the class of 
all classes <£ (a, y) as a varies. Whether we are to regard our function as 
giving classes <f> (a, y) or <f> (x, b) depends upon the order of generalization 
adopted. Thus "(g#):(3/)" involves <f>(a,y), but "(y):(^as)" involves 

Consider now the matrix <f> I x, as a function of two variables. If we first 
vary x, keeping <£ fixed (which seems the more natural order), we form a class 
of propositions <f> I x, <f> I y, <f> ! z, . . . which differ solely by the substitution of 
one individual for another. Having made one such class, we make another, 
and so on, until we have done so in all possible ways. But now suppose we 
vary <f> first, keeping x fixed and equal to a. We then first form the class of 
all propositions of the form <f> ! a, i.e. all elementary propositions of which a is 
a constituent ; we next form the class <f> I b ; and so on. The set of propositions 
which are values of <£ ! a is a set not obtainable by variation of individuals, 
i.e. not of the form fx [for constant / and variable x\ This is what makes <f> 
a new sort of variable, different from x. This also is why generalization of the 
form (<f>) . F I (<f> 1 2, x) gives a function not of the form /! x [for constant /]. 
Observe also that whereas a is a constituent of/! a, /is not ; thus the matrix 
<f> ! x has the peculiarity that, when a value is assigned to x, this value is a 
constituent of the result, but when a value is assigned to <f>, this value is 
absorbed in the resulting proposition, and completely disappears. We may 
define a function <£!& as that kind of similarity between propositions which 
exists when one results from the other by the substitution of one individual 
for another. 

We have seen that there are matrices containing, as variables, functions 
of individuals. We may denote any such matrix by 

fl(<f>lz, ^r \z,xlz, ... x,y,z, ...). 

Since a function can only occur through its values, <f> ! 2 (e.g.) can only occur 
in the above matrix through the occurrence of <f> ! x, <j> ! y, <f> ! z, . .. or of <f> I a, 
<f>lb,(f>lc, ..., where a, b, c are constants. Constants do not occur in logic, that 
is to say, the a, b, c which we have been supposing constant are to be regarded 
as obtained by an extra-logical assignment of values to variables. They may 

therefore be absorbed into the x, y, z, Now x, y, z themselves will only 

occur in logic as arguments to variable functions. Hence any matrix which 
contains the variables <f> ! z, yjr 1 2 , x • %> ®> V> z and no others, if it is of the sort 
that can occur explicitly in logic, will result from substituting <f>\x,<f>\y,$\z, 
yfrlx, yfrly, yfrlz, %lx, % 1 y, % I z, or some of them, for elementary propositions 
in some stroke-function. 



INTRODUCTION XXXI 

It is necessary here to explain what is meant when we speak of a " matrix 
that can occur explicitly in logic," or, as we may call it, a " logical matrix." 
A logical matrix is one that contains no constants. Thus p | q is a logical 
matrix ; so is <f> ! x, where <f> and x are both variable. Taking any elementary 
proposition, we shall obtain a logical matrix if we replace all its components 
and constituents by variables. Other matrices result from logical matrices by 
assigning values to some of their variables. There are, however, various ways 
of analysing a proposition, and therefore various logical matrices can be derived 
from a given proposition. Thus a proposition which is a value of p | q will 
also be a value of (<j>lx)\ (^rly) and of %!(#, y). Different forms are required 
for different purposes ; but all the forms of matrices required explicitly in 
logic are logical matrices as above denned. This is merely an illustration of 
the fact that logic aims always at complete generality. The test of a logical 
matrix is that it can be expressed without introducing any symbols other 
than those of logic, e.g. we must not require the symbol " Socrates." Consider 
the expression 

/! (<f> ! z, yfr I z, x ! z, ••• #, y, z). 
When a value is assigned to /, this represents a matrix containing the variables 

$' ty> X> • • • x > y> z > But wn il e / remains unassigned, it is a matrix of a 

new sort, containing the new variable /. We call / a " second-order function," 
because it takes functions among its arguments. When a value is assigned, 
not only to /, but also to <f>, yfr, %, . . . x t y, z, . . . , we obtain an elementary 
proposition ; but when a value is assigned to f alone, we obtain a matrix 
containing as variables only first-order functions and individuals. This is 
analogous to what happens when we consider the matrix <£ ! x. If we give 
values to both <f> and #, we obtain an elementary proposition ; but if we give 
a value to <£ alone, we obtain a matrix containing only an individual as variable. 

There is no logical matrix of the form f ! (<f> ! 2). The only matrices in 
which <f> ! 1z is the only argument are those containing <j> I a, <f> ! b, <f> ! c, . . . , where 
a, b, c, ... are constants; but these are not logical matrices, being derived 
from the logical matrix <f> \x. Since <f> can only appear through its values, it 
must appear, in a logical matrix, with one or more variable arguments. The 
simplest logical functions of <f> alone are (#) . <f> ! x and (a«) . <f> ! x, but these 
are not matrices. A logical matrix 

fl(<f)lz, a?i,# 2 , ... x n ) 
is always derived from a stroke-function 

F(pi,Pz,Ps> >..p n ) 
by substituting <p I x lt (f> ! x 2 , . . . <f> ! x n for p\, p 2> . . . p n . This is the sole method 
of constructing such matrices. (We may however have x r = x s for some values 
of r and s.) 

Second-order functions have two connected properties which first-order 
functions do not have. The first of these is that, when a value is assigned to 

R&W I c 



XXXU INTRODUCTION 

/, the result may be a logical matrix; the second is that certain constant values 
of/ can be assigned without going outside logic. 

To take the first point first:/! (<j> ! z, x), for example, is a matrix containing 
three variables,/, <£, and x. The following logical matrices (among an infinite 
number) result from the above by assigning a value to/: <f> ! x, (<j> ! x) \ (<f> ! x), 
<j>lxD<f>lx, etc. Similarly <f>lx2<f>ly, which is a logical matrix, results from 
assigning a vulue to /in/! (<£ ! 2, x, y). In all these cases, the constant value 
assigned to / is one which can be expressed in logical symbols alone (which 
was the second property of/). This is not the case with <f> ! x: in order to 
assign a value to <f>, we must introduce what we may call "empirical constants," 
such as "Socrates" and "mortality" and "being Greek." The functions of x 
that can be formed without going outside logic must involve a function as a 
generalized variable; they are (in the simplest case) such as (<f>).<f>lx and 
(a<£) .<plx. 

To some extent, however, the above peculiarity of functions of the second 
and higher orders is arbitrary. We might have adopted in logic the symbols 

Ri (x), R* {so, y), R 3 (#, y>z), 

where R± represents a variable predicate, R % a variable dyadic relation (in 
intension), and so on. Each of the symbols R x {x), R 2 (x,y), R 3 (x,y,z), ... is 
a logical matrix, so that, if we used them, we should have logical matrices not 
containing variable functions. It is perhaps worth while to remind ourselves 
of the meaning of "<f> ! a," where a is a constant. Th<^ meaning is as follows. 
Take any finite number of propositions of the various forms jRj (x), R 2 (x, y), ... 
and combine them by means of the stroke in any way desired, allowing any 
one of them to be repeated any finite number, of times. If at least one of 
them has a as a constituent, ie. is of the form 

R n (a,b 1 , b 2 , ... 6 n _j), • 
then the molecular proposition we have constructed is of the form <j> ! a, 
i.e. is a value of " <f> ! a" with a suitable <f>. This of course also holds of the 
proposition R n (a, b 1} b 2 , . . . 6 M _i) itself. It is clear that the logic of propositions, 
and still more of general propositions concerning a given argument, would be 
intolerably complicated if we abstained from the use of variable functions; 
but it can hardly be said that it would be impossible. As for the question of 
matrices, we could form a matrix/! (i2j, x), of which R t (x) would be a value. 
That is to say, the properties of second-order matrices which we have been 
discussing would also belong to matrices containing variable universals. They 
cannot belong to matrices containing only variable individuals. 

By assigning <£ ! £ and x in/! (<£ ! £, x), while leaving /variable, we obtain 
an assemblage of elementary propositions not to be obtained by means of 
variables representing individuals and first-order functions. This is why the 
new variable /is useful. 



INTRODUCTION XXX111 

We can proceed in like manner to matrices 

Fl{fl($l%$),gl($l%x), ...^\% X \$,...x,y, ...} 
and so on indefinitely. These merely represent new ways of grouping ele- 
mentary propositions, leading to new kinds of generality. 

V. FUNCTIONS OTHER THAN MATRICES 
When a matrix contains several variables, functions of some of them can 
be obtained by turning the others into apparent variables. Functions obtained 
in this way are not matrices, and their values are not elementary propositions. 
The simplest examples are 

(y) • £ '• (», V) and (ay) .<f>l(x, y). 
When we have a general proposition (<£) . F {<£ I z, x, y, ...}, the only values <f> 
can take are matrices, so that functions containing apparent variables are not 
included. We can, if we like, introduce a new variable^ to denote not only 
functions such as <f> I ot,- but also such as 

(y).<j>l($,y), (y,z).<f>l(x,y,z), ... (ay) •<£!(£, y), ...; 
in a word, all such functions of one variable as can be derived by generalization 
from matrices containing only individual-variables. Let us denote any such 
function by fax, or -ty^sc, or Xl x, or etc. Here the suffix 1 is intended to indi- 
cate that the values of the functions may be first-order propositions, resulting 
from generalization in respect of individuals. In virtue of #8, no harm can 
come from including such functions along with matrices as values of single 
variables. 

Theoretically, it is unnecessary to introduce such variables as fa, because 
they can be replaced by an infinite conjunction or disjunction. Thus e.g. 

((f),) .fax. = : (<f>). <f>lx: (fa y) .xf) ! (x, y) : (0) : (ay) .<f>l(x,y): etc., 
(a<k) . fax . = : (a<£) .<f>l x:v: (g<£) : (y) . <j> ! (x,y):v :{>&<}>, y).<f> ! (x,y) :v: etc., 
and generally, given any matrix fl(<f>lz, x), we shall have the following pro- 
cess for interpreting (c^) ./! (faz, x) and (a<£i) ./! (faz, #)• Put 

(fa) ./! (fa%x) . =. : (<f>) ./ ! {(y) .<£!(£, y), x] : (<f>) ./! {(ay) . </> ! (z, y), x], 
where/! {(y) . <f> ! (z, y), x) is constructed as follows: wherever, in/! {<£ ! z, x}, 
a value of <j>, say <f> I a, occurs, substitute (y) . <£ ! (a, y), and develop by the 
definitions at the 'beginning of #8. / ! {(ay) . <f> I (z, y), x] is similarly con- 
structed. Similarly put 

(fa) ./! (fa lz,x). = : (</>) ./! {(y, w) . <f> ! (% y, w), x) : 

(</>) -/ ! {(y) '■ (a w ) • <f> *(% y, w), x] : etc., 
where "etc." covers the prefixes (a.y) : ( w ) •> (33/> w) •> (w) : (32/)- We define 
(f> 3 , fa, ... similarly. Then 

(fa) .fl(fa% x) . = : (fa) ./! (^ 2, x) : (<£ 2 ) ./! (fa 3, x) : etc. 
This process depends upon the fact that/! (<£ ! z, x), for each value of <}> and x, 
is a proposition constructed out of elementary propositions by the stroke, and 

c2 



XXxiv INTRODUCTION 

that #8 enables us to replace any of these by a proposition which is not 
elementary. (a<£i) .flifa'z, x) is defined by an exactly analogous disjunction. 

It is obvious that, in practice, an infinite conjunction or disjunction such 
as the above cannot be manipulated without assumptions ad hoc. We can 
work out results for any segment of the infinite conjunction or disjunction, 
and we can " see " that these results hold throughout. But we cannot prove 
this, because mathematical induction is not applicable. We therefore adopt 
certain primitive propositions, which assert only that what we can prove in 
each case holds generally. By means of these it becomes possible to manipulate 
such variables as fa. 

In like manner we can introduce /, (faz, £), where any number of in- 
dividuals and functions yjr 1} ft, ... may appear as apparent variables. 

No essential difficulty arises in this process so long as the apparent 
variables involved in a function are not of higher order than the argument to 
the function. For example, x e D'JR, which is (ay) . xRy, may be treated 
without danger as if it were of the form <f> ! x. In virtue of #8, fax may be 
substituted for <£ ! x without interfering with the truth of any logical pro- 
position which <f> ! x is a part. Similarly whatever logical proposition holds 
concerning/! (faz, x) will hold concerning f x (faz, x). 

But when the apparent variable is of higher order than the argument, a 
new situation arises. The simplest cases are 

(*)./! ($!*,*), (3*) ■/! (*!*,*). 
These are functions of x, but are obviously not included among the values 
for" <f> ! x (where <f> is the argument). If we adopt a new variable fa which is 
to include functions in which (f> ! z can be an apparent variable, we shall obtain 
other new functions 

ifa).f\{fa%x), (afc) ./!(#*,*)> 

which are again not among values for fax (where fa is the argument), because 
the totality of values of faz, which is now involved, is different from the totality 
of values of <f> ! £, which was formerly involved. However much we may en- 
large the meaning of <f>, a function of x in which <f> occurs as apparent variable 
has a correspondingly enlarged meaning, so that, however <f> may be defined, 

(fa).f\(4>%x) and (a*) ./!(#,*) 
can never be values for <f>x. To attempt to make them so is like attempting 
to catch one's own shadow. It is impossible to obtain one variable which 
embraces among its values all possible functions of individuals. 

We denote by fax a function of x in which fa is an apparent variable, but 
there is no variable of higher order. Similarly fax will contain fa as apparent 
variable, and so on. 



INTRODUCTION XXXV 

The essence of the matter is that -a variable may travel through any well- 
defined totality of values, provided these values are all such that any one can 
replace any other significantly in any context. In constructing fax, the only 
totality involved is that of individuals, which is already presupposed. But 
when we allow <j> to be an apparent variable in a function of x, we enlarge the 
totality of functions of a;, however <f> may have been defined. It is therefore 
always necessary to specify what sort of <j> is involved, whenever <f> appears as 
an apparent variable. 

The other condition, that of significance, is fully provided for by the 
definitions of *8, together with the principle that a function can only occur 
through its values. In virtue of the principle, a function of a function is a 
stroke-function of values of the function. And in virtue of the definitions in 
*8, a value of any function can significantly replace any proposition in a 
stroke-function, because propositions containing any number of apparent 
variables can always be substituted for elementary propositions and for each 
other in any stroke-function. What is necessary for significance is that every 
complete asserted proposition should be derived from a matrix by generaliza- 
tion, and that, in the matrix, the substitution of constant values for the 
variables should always result, ultimately, in a stroke-function of atomic 
propositions. We say " ultimately," because, when such variables as fa% are 
admitted, the substitution of a value for fa may yield a proposition still 
containing apparent variables, and in this proposition the apparent variables 
must be replaced by constants before we arrive at a stroke-function of atomic 
propositions. We may introduce variables requiring several such stages, but 
the end must always be the same : a stroke-function of atomic propositions. 

It seems, however, though it might be difficult to prove formally, that the 
functions fa, fi introduce no propositions that cannot be expressed without 
them. Let us take first a very simple illustration. Consider the proposition 

(H^i) ■ fa x m fa a > which we w *^ call /(a?, a). 
Since fa includes all possible values of <f> ! and also a great many-other values 
in its range, /(«, a) might seem to make a smaller assertion than would be 

made by 

(g<£) . <f> I x . <j> ! a, which we will call/, (x, a). 

But in fact f{x, a) . D ./„ (x, a). This may be seen as follows : fax has one of 

the various sets of forms : 

(y) . 4> ! (x, y), (y, z).<}>l 0, y, z), ..., 

(ay) ■ $ ■ 0*> y). to *) • tf> ! fo y.*).—> 

(y) : (a*) • <M 0»» y. *)> (ay) : (*) • ! fo-y» z ^ 

Suppose first that fax . = . (y) . <f> ! (x, y). Then 



fax . faa . = 

D 



(y) . <j> I (x, y) : (y) . tf> ! (a, y) 
<f, I (x, b).<f>l (a, b) : 
(a</>) . <£ I x . <£ ! a. 



XXXVI INTRODUCTION 

Next suppose fax . = . fay) .<f>\{x, y). Then 

fax .faa. = : (gy) . <f> ! O, ?/) : faz) .<f>l(a, z) : 

3 '■ (ay, z):<f>l(oe ) y)v<f>l (x, z).fa\ (a, y)v<f>l (a, z) : 
D : (g;0) . <j> I x . <j> ! a, 
because <j> I (%, y) v (f> 1 (x, z) is of the form <f> I x, when y and z are fixed. It is 
obvious that this method of proof applies to the other cases mentioned above. 
Hence 

fafa) . fax . faa . = . (>&<j>) . <f> 1 x . <f> I a. 

We can satisfy ourselves that the same result holds in the general form 

(a&)./! (<M>*) ■ = ■ (a*) -/! (*!*,-*) 

by a similar argument. We know that / ! (0 ! £, a?) is derived from some 
stroke-function 

F(p,q,r,...) 
by substituting <f> I x, <f> I a, </> ! b, . . . (where a, b, ... are constants) for some of 
the propositions p,q,r,... and g x l x, g 2 lx, g 3 lx, ... (where ^, # 2 , g s , ... are 
constants) for others of p, q, r, ..., while replacing any remaining propositions 
p, q, r, ... by constant propositions. Take a typical case ; suppose 

fl(<l>lz,x). = .(<f>la)\{(<f>lx)\(cl>lb)}. 
We then have to prove 

faa\(fax\fab).D.fa<f>).<f>la\(falx\<f>lb), 
where fax may have any of the forms enumerated above. 

Suppose first that fax . — . (y) . <$> ! (x, y). Then 

faa | (fax | fab) . = : (ay) :(z,w).<f>\ (a, y)\{<f>l (x, z)\<t>\ (b, w)} : 
D : (32/) . fal (a, y) \ {<f> ! (x, y)\<f>l (b, y)} : 
D:( a <£).<£!a|(<£!tf|0!&) 
because, for a given y, <f> ! (x, y) is of the form <f> I x. 

Suppose next that fax . = . (33/) . <j> ! (x, y). Then 
faa I (&« J fab) . = : (y) : faz, w).<f>l (a, y) | {<f> ! (a;, *) | <f> ! (6, w)} : 

D : (a>|r) . yjr ! a j (^ ! x | i/r ! b), 
putting \jrlx .= . (ftl(x,z)v<j>l(x, w). Similarly the other cases can be dealt 
with. Hence the result follows. 

Consider next the correlative proposition 

(fa) ./! (fa% x) . = . (<£) ./! (<£ ! X x). 
Here it is the converse implication that needs proving, i.e. 

(fa).f\(<t>l%x).1.(fa).f\(fa%x). 
This follows from the previous case by transposition. It can also be seen in- 
dependently as follows. Suppose, as before, that 

fl(fa$,x). = .(faa)\(fax\fab), 
and put first fax . = . (y) .<f>\(x, y). 

Then (faa) \ (fax \ fab) . = : ( H y) : (z, w).<f>\ (a, y)\{<f>l (x, z)\<f>l (6, «/)}. 



INTRODUCTION XXXV11 

Thus we require that, given 

(ylr).(ylrla)\(yfrlx\^lb), 
we should have (g#) : (z, w) . <f> I (a, y)\{<f>l (x, z) \ <j> ! (6, w)}. 
Now 
(yft) . yfr ! a \ (yfr ! x \ yfr ! b) . D : . <f> ! (a, z) . D . <f> ! (#, z) . <f> I (b, z) : 

<f> ! (a, w) . D . <£ ! (x, w) . <f> ! (6, «/) :. 

D :. <f> ! (a, *) . <£ ! (a, w) . D . <£ ! (x, z).<f>l (b, w) :. 

D:.<f>l(a,w).D:<f>l(a,z).D.(j>l(x > z).(l>l(b > w) (1) 
Also ~^>!(a,?«).D:<^!(a,w;).D.</>!(«,5).^!(6 ) w) (2) 

(l).(2).D:.(^).^!o|(^!ar|^!6):D:.(ay):^!(a,y).D.^!(a?,«).^!(6,w) 
which was to be proved. 

Put next fax . = . (33/) . <£ ! (x, y). 

Then (fca) | (fax \ fab). = :(y): faz, w).<f>l (a, y) | {</> ! (x, z) \ <f> ! (6, w)}. 
In this case we merely put z = w = y and the result follows. 

The method will be the same in any other case. Hence generally : 
(fa) ./! (fa% x). = , (<j>) .fl(<j> \X x). 
Although the above arguments do not amount to formal proofs, they suffice 
to make it clear that, in fact, any general propositions about <j> ! z are also 
true about faz. This gives us, so far as such functions are concerned, all that 
could have been got from the axiom of reducibility. 

Since the proof can only be conducted in each separate case, it is necessary 
to introduce a primitive proposition stating that the result holds always. This 
primitive proposition is 

h :(*)./! (01 % x).D.fl(fa%x) Pp. 
As an illustration : suppose we have proved some property of all classes denned 
by functions of the form <f> ! z, the above primitive proposition enables us to 
substitute the class T)'R, where R is the relation denned by <f> ! (x, p), or by 
(gs) . <f> ! (x, $, z), or etc. Wherever a class or relation is denned by a function 
containing no apparent variables except individuals, the above primitive pro- 
position enables us to treat it as if it were denned by a matrix. 

We have nOw to consider functions of the form fax, where 

fax . = . (<£) ./! (<f> I % x) or fax . = . (gtf) ./! (<f> I % x). 
We want to discover whether, or under what circumstances, we have 

(fa) .g\(4>\^x) .1 . g\(faz,x). (A) 

Let us begin with an important particular case. Put 
gl(<f>lz,x). = .<f>laD<f>lx. 
Then (fa . g I (<f> I z, x) . = . x = a, according to #131. 



XXXV111 INTRODUCTION 

We want to prove 

(<j>) . <j> I a D <f> I x . D . <f> 2 a D <f> 2 x, 
i.e. (<f>).<t>laD<f>lx.3: (£) ./! (tf> ! z, a) . D . (<f>) ./!(</>! % x) : 

(a<*>) •/! (<*> l%a).D. ( a <£) ./!(<£! *, *). 
Now/! (0"! 2, a?) must be derived from some stroke-function 

F(p,q,r,...) 

by substituting for some of p, q,r, ... the values <j> I x, tf> I b, ! c, . . . where 

b, c, ... are constants. As soon as <f> is assigned, this is of the form yfr ! #. Hence 

(<f>).(j>laD<f>lx.D :(<!>) :/! (<£ ! % a) . D ./! (<f> ! % x) : 

D:(*)./!(*!*,a). 3. (*)./!(*!*,*): 

(a<*>) ■/'■ (<*> I % a) • ^ • (3*) ■/!■(* ! *, *)• 
Thus generally (<£) . </> ! a D <£ ! x . D . (<£ 2 ) . <f> 2 a D </> 2 a? without the need of any 
axiom of.reducibility. 

It must not, however, be assumed that (A) is always true. The procedure 
is as follows :/!(</>! 2, x) results from some stroke-function 

F(p,q,r,...) 
by substituting for some of p, q,r, ... the values <£ ! x, <j> ! a, <f> I b, ... (a, b, ... 
being constants). We assume that, e.g. 

4> 2 x. = .{<j>).f\{4>\z,x). 
Thus <f, 2 x. = .(<}>). F((j> I x, <j>la, <f>lb, ...). (B) 

What we want to discover is whether 

{<\>).g\{^\%x).^.g\{^%x). 
Now g ! (<f> I z, x) will be derived from a stroke-function 

G(p,q,r,...) 

by substituting <f> I x, <j>la', <f>lb', ... for some of p, q, r, To obtain 

g\($ 2 z,%), we have to put <f> 2 x, <f> 2 a, <f> 2 b', ... in G(p, q, r, ...), instead of 
<f> ! x, <f>la', <f>lb', We shall thus obtain a new matrix. 

If ((f>) . g I ((f) ! z, x) is known to be true because G(p, q, r, ...) is always 
true, then g ! (<f> 2 z, x) is true in virtue of #8, because it is obtained from 
G (p, q, r, ...) by substituting for some of p, q, r, ... the propositions <f> 2 x, 
<f> 2 a', <f> 2 b', ... which contain apparent variables. Thus in this case an inference 
is warranted. 

We have thus the following important proposition : 

Whenever (</>) . gl(<j>lz,x) is known to be true because g ! (<£ ! z,x) is 
always a value of a stroke-function 

G(p, q, r, ...), 
which is true for all values of p, q, r, ..., then g ! (<f> 2 lz, x) is also true, and so 
(of course) is (<£ 2 ) . g ! (<f> 2 z. x). 



INTRODUCTION XXXIX 

This, however, does not cover the case where (<j>) . g ! (<f> ! 2, x) is not a 
truth of logic, but a hypothesis, which may be true for some values of x and 
false for others. When this is the case, the infereDce to g ! (<£ 2 2, x) is some- 
times legitimate and sometimes not ; the various cases must be investigated 
separately. We shall have an important illustration of the failure of the 
inference in connection with mathematical induction. 

VI. CLASSES 

The theory of classes is at once simplified in one direction and complicated 
in another by the assumption that functions only occur through their values 
and by the abandonment of the axiom of reducibility. 

According to our present theory, all functions of functions are extensional, 



i.e. 



<t>x= x +x.l.f(p)=f(1rt). 

This is obvious, since <f> can only occur in f(4>z) by the substitution of values 
of <£ for p, q, r, ... in a stroke-function, and, if <f>x = yfrx, the substitution of 
§x for p in a stroke-function gives the same truth-value to the truth-function 
as the substitution of yfrx. Consequently there is no longer any reason to 
distinguish between functions and classes, for we have, in virtue of the above, 

<j)x = x tyx . D . <f>% = yjrx. 

We shall continue to use the notation & (<$>x), which is often more convenient 
than <j)tc ; but there will no longer be any difference between the meanings of 
the two symbols. Thus classes, as distinct from functions, lose even that 
shadowy being which they retain in #20. The same, of course, applies to 
relations in extension. This, so far, is a simplification. 

On the other hand, we now have to distinguish classes of different orders 
composed of members of the same order. Taking classes of individuals as the 
simplest case, & (<£> ! x) must be distinguished from & (<f> 2 x) and so on. In 
virtue of the proposition at the end of the last section, the general logical 
properties of classes will be the same for classes of all orders. Thus e.g. 

aC/3./3C7.D.aC 7 

will hold whatever may be the orders of a, #, y respectively. In other kinds of 
cases, however, trouble arises. Take, as a first instance, p l K and s'k. We have 

x ep f K . = : a e k . D„ . x e a. 
Thus p'tc is a class of higher order than any of the members of k. Hence the 
hypothesis (a) .fa may not imply f{p'ic), if a is of the order of the members 
of k. There is a kind of proof invented by Zermelo, of which the simplest 
example is his second proof of the Schroder-Bernstein theorem (given in #73). 
This kind of proof consists in defining a certain class of classes tc, and then 
showing that p'tceic. On the face of it, "p'/ce/c" is impossible, since p'/e is 



Xl INTRODUCTION 

not of the same order as members of k. This, however, is not all that is to be 
said. A class of classes k is always denned by some function of the form 

Ox, x 2 , ...): (gy x , y 2 , . . .) . Ffa e a, x 2 e a, . . . y x e a, y 2 e a, . . .), 

where F is a stroke-function, and "oe«" means that the above function is 
true. It may well happen that the above function is true when p'ic is sub- 
stituted for a, and the result is interpreted by #8. Does this justify us in 
asserting p*K etc? 

Let us take an illustration which is important in connection with 
mathematical induction. Put 

K = a (R"a Ca.aea). 

Then R"p'/cCp'K . aep'/c (see *40'81) 

so that, in a sense, p i K e k. That is to say, if we substitute p l K for a in the 
defining function of k, and apply #8, we obtain a true proposition. By the 
definition of #90, 

4— 

R%. t a=p t K. 
<— 

Thus R%a is a second-order class. Consequently, if we have a hypothesis 

(a) .fa, where a is a first-order class, we cannot assume 

(a)./a.D./CR*'a). (A) 

By the proposition at the end of the previous section, if (a) ./a is deduced by 
logic from a universally-true stroke-function of elementary propositions, 

f(R%a) will also be true. Thus we may substitute R#a for a in any asserted 
proposition " h .fa" which occurs in Principia Mathematica. But when 
(a) ./a is a hypothesis, not a universal truth, the implication (A) is not, prima 
facie, necessarily true. 

For example, if k = a (R"a C a . a e a), we have 

ae*.D:a«/3e/c. = . R"(a n@)C@ .ae/3. 

Hence a e k . R"{ar\ 0) C /3 . a e . D .p'ic C /3 (1) 

In many of the propositions of #90, as hitherto proved, we substitute p'ic for 
a, whence we obtain 

R"(/3np<,e)C/3.ae/3.D.p t feC/3 (2) 

i.e. 

z e . aR%z . D Z) w . w e y8 : a e . aR%oc : D . x e /8 

or aR^x . D :. z e /3 . aR%z . D z w .M/e/3:ae/8:D.#e/3. 

This is a more powerful form of induction than that used in the definition of 
aR%x. But the proof is not valid, because we have no right to substitute p'tc 
for a in passing from (1) to (2). Therefore the proofs which use this form of 
induction have to be reconstructed. 



INTRODUCTION xli 

It will be found that the form to which we can reduce most of the fallacious 
inferences that seem plausible is the following: 

Given " h . (x) . f(x, %)" we can infer " h : (x) : (gy) . f(x, y)." Thus given 
" I- . (a) ./(a, a)" we can infer " V : (a) : (g£) ./(a, #)." But this depends upon 
the possibility of a = 0. If, now, a is of one order and /8 of another, we do 
not know that a = /? is possible. Thus suppose we have 

a e k . D a . got. 
and we wish to infer g$, where # is a class of higher order satisfying /3 e k. 
The proposition 

(/3) :. a e /e . D a . #a : D : /3 e /c . D . gryS 

becomes, when developed by #8, 

(£) :: (ga) :.ae re .3 .ga-.D : fie k .D .g/3. 
This is only valid if o = $ is possible. Hence the inference is fallacious if /3 
is of higher order than a. 

Let us apply these considerations to Zermelo's proof of the Schroder- 
Bernstein theorem, given in *73"8 ff. We have a class of classes 

* = 3(a C D'R . £- <l l R C a . R"a Ca) 
and we prove p'tc e x (#73"81), which is admissible in the limited sense ex- 
plained above. We then add the hypothesis 

x~e(0-Q.'R)vIt"p t ic 

and proceed to prove p l K — i'x e k (in the fourth line of the proof of *7382). 
This also is admissible in the limited sense. But in the next line of the same 
proof we make a use of it which is not admissible, arguing from p*K — i'xe k 
to p'x Cp'tc — i l x, because 

ae k . D a . p'k C a. 
The inference from 

a e k . D a .p ( K Ca to p'/c— t'xe k .0 . p'/cCp'ic — i'x 
is only valid if p e /c — i'x is a class of the same order as the members of k. 
For, when a e k . D a . p l K C a is written out it becomes 

(a) ::: (g/S) ::. (x) :: a e k . D :. ft e k . D .xe /S : D . x e a. 

This is deduced from 

a e k . O :. a e k . D . x e a : D . x e a 

by the principle that /(a, a) implies (g/3) ./(a, /3). But here the fi must be 
of the same order as the a, while in our case a and /8 are not of the same 
order, if a = p l K — i l x and /3 is an ordinary member of k. At this point, there- 
fore, where we infer p l K Qp'ic — fc'#,*the proof breaks down. 

It is easy, however, to remedy this defect in the proof. All we need is 

x~e(@-<l'R)u R"p'/c. D.x~ep'/c 
or, conversely, 



Xlii INTRODUCTION 

Now 

x ep'/c . D :. a e k . D a : a — i'x^e k : 

3 a . ^(p _ a f jR C a - i'a>) . v . ~ [R"(ol - i'x) C a - t'«?} : 
X :xe0- Q.'R . v . xe R"{<t- l'x) 

D :. x e - CF# : v : a e k . D a . x e i£"a. 
Hence, by *72-341, 

a; e|><* . D . x e (J3 - d'R) u R"p<" 
which gives the required result. 

We assume that a — l'x is of no higher order than a; this can be secured 
by taking a to be of at least the second order, since i'x, and therefore — t'oc, 
is of the second order. We may always assume our classes raised to a given 
order, but not raised indefinitely. 

Thus the Schroder-Bernstein theorem survives. 

Another difficulty arises in regard to sub-classes. We put 

Cl'a = /§(/3Ca) Df. 

Now "0Ca" is significant when /3 is of higher order than a, provided its 

members are of the same type as those of a. But when we have 

0Ca.Dp.ffr 

the /3 must be of some definite type. As a rule, we shall be able to show 

that a proposition of this sort holds whatever the type of /3, if we can show 

that it holds when is of the same type as a. Consequently no difficulty 

arises until we come to Cantor's proposition 2 W > n, which results from the 

proposition 

~{(Cl'a)sm«} 

which is proved in #102. The proof is as follows: 

R e 1 -» 1 . WR = a . <J'i2 C Cl'a . £ = x [x e a - R'x) . D : 
^ w ^ 

ye a. ye R'y .Oy.y^eg-.yeu. y~e R'y .D y .ye^:D:yea.D y .^ R'y : 

D :£<-€<!<#. 
As this proposition is crucial, we shall enter into it somewhat minutely. 

Let a = £ (A ! x), and let 

xR${4>\z)). = .f\{4>\%x). 

Then by our data, 

Alx.D. (?[<!>). fl(<j>l%x), 

fl(<f>l^,x).D.Alx.cf>lyD y Aly > 
fl(<f>lz,x).fl(<l>lz,y).O.x = y, 
f ! (<f> I % x).f\{^r\%x).0.<i>\y =y.^ ! y. 
With these data, 

xea — R'x . = '. A ! x :/!(<£! z, x) . D^ . ~ <j> I x. 

Thus £ = £{(<£) '• A '■ x : / ! (4> '■ 2. x ) ■ D -~4> '■ x )- 



INTRODUCTION xliii 

Thus £ is defined by a function in which <\> appears as apparent variable. If 
we enlarge the initial range of <f>, we shall enlarge the range of values involved 
in the definition of £. There is therefore no. way of escaping from the result 
that £ is of higher order than the sub-classes of a contemplated in the 
definition of Cl'a. Consequently the proof of 2 n >• n collapses when the 
axiom of reducibility is not assumed. We shall find, however, that the propo- 
sition remains true when n is finite. 

With regard to relations, exactly similar questions arise as with regard to 
classes. A relation is no longer to be distinguished from a function of two 
variables, and we have 

0(£,£) = ^(£,#) . = :<f>{x,y) . =,,„ .f(x,y). 
The difficulties as regards^'X and Rl'Pare less important than those concerning 
p'/c and Cl'a, because p l \ and HYP are less used. But a very serious difficulty 
occurs as regards similarity. We have 

a sm . = . (rR) . i2 e 1 -* 1 . a = D'i£ . /3 = (Pi*. 
Here R must be confined within some type; but whatever type we choose, 
there may be a correlator of higher type by which o and can be correlated. 
Thus we can never prove <~(asmy8), except in such special cases as when 
either a or is finite. This difficulty was illustrated by Cantor's theorem 
2 n > n, which we have just examined. Almost all our propositions are con- 
cerned in proving that two classes are similar, and these can all be interpreted 
so as to remain valid. But the few propositions which are concerned with 
proving that two classes are not similar collapse, except where one at least of 
the two is finite. 

VII. MATHEMATICAL INDUCTION 

All the propositions on mathematical induction in Part II, Section E and 
Part III, Section C remain valid, when suitably interpreted. But the proofs 
of many of them become fallacious when the axiom of reducibility is not 
assumed, and in some cases new proofs can only be obtained with considerable 
labour. The difficulty becomes at once apparent on observing the definition 
of "xR%y" in #90. Omitting the factor "xeC'R," which is irrelevant for 
our purposes, the definition of " xR%y " may be written 

zRw.'D z>w .<l>lz'2 4>l w.D^.^lxD^ly, (A) 

i.e. " y has every elementary hereditary property possessed by x." We may, 
instead of elementary properties, take any other order of properties ; as we 
shall see later, it is advantageous to take third-order properties when R is 
one-many or many- one, and fifth-order properties in other cases. But for 
preliminary purposes it makes no difference what order of properties we take, 
and therefore for the sake of definiteness we take elementary properties to 
begin with. The difficulty is that, if <f> 2 is any second-order property, we 
cannot deduce from (A) 

zRw . D z>w . <j> 2 z D <p 2 w : D . <f> 2 x D <f> 2 y. (B) 



xliv 



INTRODUCTION 



Suppose, for example, that <j> 2 z.= .(<f>).fl(<frlz,z); then from (A) we can 

deduce 

zRw . D Zj w ./! (<£ ! % z) D 4 /! (£ ! % iv) : D :/! (<£ ! % x) . D* ./! (<£ ! £ , y) : 

D : fax . D . <j> 2 y. (C) 

But in general our hypothesis here is not implied by the hypothesis of.(B). 
If we put <f> 2 z . = . (g<£) .f\ {<f> I z, z), we get exactly analogous results. 

Hence in order to apply mathematical induction to a second-order property, 
it is not sufficient that it should be itself hereditary, but it must be composed 
of hereditary elementary properties. That is to say, if the property in question 
is <f> 2 z, where (f> 2 z is either 

(<f>) ./! (* ! % z) or (a*) ./! <* ! *,*), 
it is not enough to have 

zRw .D 2>w .<l> 2 z~)<f> 2 w, 

but we must have, for each elementary 0, 

zRw.D ZtW ./l t4> ! % z) D/I (0 ! f , «;). 

■ One inconvenient consequence is that, primd facie, an inductive property 
must not be of the form 

xR%. z . <f>lz 

or SeFotid'R.tfilS 

or a e NC induct . <}> I a. 

This is inconvenient, because often such properties are hereditary when <f> 

alone is not, i.e. we may have 

xR^z .<f>lz .zRw . D 2(OT . xR% w . <ft ! w 
when we do not have 

<f> ! z . zRw . D z>w .<j>lw, 
and similarly in the other cases. 

These considerations make it necessary to re-examine all inductive proofs. 
In some cases they are still valid, in others they are easily rectified; in still 
others, the rectification is laborious, but it is always possible. The method of 
rectification is explained in Appendix B to this volume. 

There is, however, so far as we can discover, no way by which our present 
primitive propositions can be made adequate to Dedekindian and well-ordered 
relations. The practical uses of Dedekindian relations depend upon #211 "63 — 
•692, which lead to #214'3 — '34, showing that the series of segments of a series 
is Dedekindian. It is upon this that the theory of real numbers rests, real 
numbers being defined as segments of the series of rationals. This subject is 
dealt with in #310. If we were to regard as doubtful the proposition that the 
series of real numbers is Dedekindian, analysis would collapse. 

The proofs of this proposition in Principia Mathematica depend upon the 
axiom of reducibility, since they depend upon #21 1*64, which asserts 

XCD'Pe.D.s'XeD'Pe. 



INTRODUCTION xlv 

For reasons explained above, if a is of the order of members of X, (a) .fa may 
not imply f(s'\), because s'\ is a class of higher order than the members of 
\. Thus although we have 

D<P e = S{( a #).a = P"/3}, 

s<\ = P"s'Pe"K 

yet we cannot infer s'X e D f P € except when s*\ or s'P e "\ is, for some special 
reason, of the same order as the members of X. This will be the case when X 
is finite, but not necessarily otherwise. Hence the theory of irrationals will 
require reconstruction. 

Exactly similar difficulties arise in regard to well-ordered series. The 
theory of well-ordered series rests on the definition #250*01 : 

Bord^PCClex'C'PCCFminp) Df, 
whence PeBord . = : aCC'P.g ! a . D a .g ! a-P"a. 

In making deductions, we constantly substitute for a some constructed class 
of higher order than C'P. For instance, in #250*122 we substitute for a the 

class G l P r\p'P"(a r\ C'P), which is in general of higher order than a. If this 
substitution is illegitimate, we cannot prove that a class contained in C'P 
and having successors must have an immediate successor, without which the 
theory of well-ordered series becomes impossible. This particular difficulty 
might be overcome, but it is obvious that many important propositions must 
collapse. 

It might be possible to sacrifice infinite well-ordered series to logical 
rigour, but the theory of real numbers is an integral part of ordinary mathe- 
matics, and can hardly be the object of a reasonable doubt. We are therefore 
justified in supposing that some logical axiom which is true will justify it. 
The axiom required may be more restricted than the axiom of reducibility, 
but, if so, it remains to be discovered. 

The following are among the contributions to mathematical logic since the 
publication of the first edition of Principia Mathematica. 

D. Hilbert. Axiomatisches Denken, Mathematische Annalen, Vol. 78. Die logischen 
Grundlagen der Mathematik, ib. Vol. 88. Neue Begriindung der Mathematik, 
Abhandlungen aus dem mathematischen Seminar der Hamburgischen UniversitcU, 1922. 

P. Bernays. Ueber Hilbert's Gedanken zur {jrundleguug der Arithmetik, Jahresbei-icht 
der deutschen Mathematiker- Vereinigung, Vol. 31. 

H. Behmanx. Beitrage zur Algebra der Logik. Mathematische Annalen, Vol. 86. 

L. Chwistek. Ueber die Antinomien der Prinzipien der Mathematik, Mathematische 
Zeitschrift, Vol. 14. The Theory of Constructive Types. Annates de la Societe 
Mathematique de Pologne, 1923. (Dr Chwistek has kindly allowed us to read in MS. 
a longer work with the same title.) 

H. Weyl. Das Kontinuum, Veit, 1918. Ueber die neue Grundlagenkrise der Mathematik, 
Mat/iematische Zeitschrift, Vol. 10. Bandbemerkungen zu Hauptproblemen der 
Mathematik, Mathematische Zeitschrift, Vol. 20. 



Xlvi INTRODUCTION 

L. E. J. Brouwer. Begriindung der Mengenlehre unabhangig vom logischen Satz des 
ausgeschlossenen Dritten. Yerhandelingen d. K. Akademie v. Wetenschappen, Amster- 
dam, 1918, 1919. Intuitionistische Mengenlehre, Jahresbericht der deutschen Mathema- 
tiker- Vereinigung, Vol. 28. 

A. Tajtelbaum-Tarski. Sur le terme primitif de la logistique, Fundamenta Mathematical, 
Tom. IV. Sur les "truth-functions" au sens de MM. Russell et Whitehead, ib. 
Tom. V. Sur quelques theoremes qui equivalent a l'axiome du choix, ib. 

F. Bernstein. Die Mengenlehre Georg Cantor's und der Finitismus, Jahresbericht der 
deutschen Mathematiker- Vereinigung, Vol. 28. 

J. Konig. Neue Orundlagen der Logik, Arithmetik und Mengenlehre, Veit, 1914. 

C. I. Lewis. A Survey of Symbolic Logic, University of California, 1918. 

H. M. Shefper. Total determinations of deductive systems with special reference to the 
Algebra of Logic. Bulletin of the American Mathematical Society, Vol. xvi. Trans. Amer. 
Math. Soc. Vol. xiv. pp. 481—488. The general theory of notational relativity, Cam- 
bridge, Mass. 1921. 

J. G. P. Nigod. A reduction in the number of the primitive propositions of logic. Proc. 
Camb. Phil. Soc. Vol. xix. 

L. Wittgenstein. Tractatus Logico-Philosophicus, Kegan Paul, 1922. 

M. Schonwinkel. Ueber die Bausteine der mathematischen Logik, Math. Annalen, Vol. 92. 



INTRODUCTION 

The mathematical logic which occupies Part I of the present work has 
been constructed under the guidance of three different purposes. In the first 
place, it aims at effecting the greatest possible analysis of the ideas with 
which it deals and of the processes by which it conducts demonstrations, 
and at diminishing to the utmost the number of the undefined ideas and 
undemonstrated propositions (called respectively primitive ideas and primitive 
propositions) from which it starts. In the second place, it is framed with a 
view to the perfectly precise expression, in its symbols, of mathematical 
propositions : to secure such expression, and to secure it in the simplest and 
most convenient notation possible, is the chief motive in the choice of topics. 
In the third place, the system is specially framed to solve the paradoxes 
which, in recent years, have troubled students of symbolic logic and the 
theory of aggregates ; it is believed that the theory of types, as set forth in 
what follows, leads both to the avoidance of contradictions, and to the 
detection of the precise fallacy which has given rise to them. 

Of the above three purposes, the first and third often compel us to adopt 
methods, definitions, and notations which are more complicated or more 
difficult than they would be if we had the second object alone in view. This 
applies especially to the theory of descriptive expressions (#14 and #30) and 
to the theory of classes and relations (#20 and #21). On these two points, 
and to a lesser degree on others, it has been found necessary to make some 
sacrifice of lucidity to correctness. The sacrifice is, however, in the main 
only temporary : in each case, the notation ultimately adopted, though its 
real meaning is very complicated, has an apparently simple meaning which, 
except at certain crucial points, can without danger be substituted in 
thought for the real meaning. It is therefore convenient, in a preliminary 
explanation of the notation, to treat these apparently simple meanings as 
primitive ideas, i.e. as ideas introduced without definition. When the notation 
has grown more or less familiar, it is easier to follow the more complicated 
explanations which we believe to be more correct. In the body of the work, 
where it is necessary to adhere rigidly to the strict logical order, the easier 
order of development could not be adopted ; it is therefore given in the 
Introduction. The explanations given in Chapter I of the Introduction are 
such as place lucidity before correctness ; the full explanations are partly 
supplied in succeeding Chapters of the Introduction, partly given in the body 
of the work. 

The use of a symbolism, other than that of words, in all parts of the book 
which aim at embodying strictly accurate demonstrative reasoning, has been 
r&w i 1 



2 INTRODUCTION 

forced on us by the consistent pursuit of the above three purposes. The 
reasons for this extension of symbolism beyond the familiar regions of number 
and allied ideas are manjr : 

(1) The ideas here employed are more abstract than those familiarly con- 
sidered in language. Accordingly there are no words which are used mainly 
in the exact consistent senses which are required here. Any use of words 
would require unnatural limitations to their ordinary meanings, which would 
be in fact more difficult to remember consistently than are the definitions of 
entirely new symbols. 

(2) The grammatical structure of language is adapted to a wide variety 
of usages. Thus it possesses no unique simplicity in representing the few 
simple, though highly abstract, processes and ideas arising in the deductive 
trains of reasoning employed here. In fact the very abstract simplicity of the 
ideas of this work defeats language. Language can represent complex ideas 
more easily. The proposition " a whale is big " represents language at its best, 
giving terse expression to a complicated fact ; while the true analysis of " one 
is a number" leads, in language, to an intolerable prolixity. Accordingly 
terseness is gained by using a symbolism especially designed to represent the 
ideas and processes of deduction which occur in this work. 

(3) The adaptation of the rules of the symbolism to the processes of 
deduction aids the intuition in regions too abstract for the imagination 
readily to present to the mind the true relation between the ideas employed. 
For various collocations of symbols become familiar as representing im- 
portant collocations of ideas ; and in turn the possible relations — according 
to the rules of the symbolism — between these collocations of symbols become 
familiar, and these further collocations represent still more complicated 
relations between the abstract ideas. And thus the mind is finally led to 
construct trains of reasoning in regions of thought in which the imagination 
would be entirely unable to sustain itself without symbolic help. Ordinary 
language yields no such help. Its grammatical structure does not represent 
uniquely the relations between the ideas involved. Thus, " a whale is big " 
and " one is a number " both look alike, so that the eye gives no help to the 
imagination. 

(4) The terseness of the symbolism enables a whole proposition to be 
represented to the eyesight as one whole, or at most in two or three parts 
divided where the natural breaks, represented in the symbolism, occur. This 
is a humble property, but is in fact very important in connection with the 
advantages enumerated under the heading (3). 

(5) .The attainment of the first- mentioned object of this work, namely 
the complete enumeration of all the ideas and steps, in reasoning employed 



INTRODUCTION O 

in mathematics, necessitates both terseness and the presentation of each pro- 
position with the maximum of formality in a form as characteristic of itself 
as possible. 

Further light on the methods and symbolism of this book is thrown by a 
slight consideration of the limits to their useful employment : 

(a) Most mathematical investigation is concerned not with the analysis 
of the complete process of reasoning, but with the presentation of such an 
abstract of the proof as is sufficient to convince a properly instructed mind. 
For such investigations the detailed presentation of the steps in reasoning is 
of course unnecessary, provided that the detail is carried far enough to guard 
against error. In this connection it may be remembered that the investiga- 
tions of Weierstrass and others of the same school have shown that, even in 
the common topics of mathematical thought, much more detail is necessary 
than previous generations of mathematicians had anticipated. 

(/3) In proportion as the imagination works easily in any region of 
thought, symbolism (except for the express purpose of analysis) becomes only 
necessary as a convenient shorthand writing to register results obtained 
without its help. It is a subsidiary object of this work to show that, with 
the aid of symbolism, deductive reasoning can be extended to regions of 
thought not usually supposed amenable to mathematical treatment. And 
until the ideas of such branches of knowledge have become more familiar, 
the detailed type of reasoning, which is also required for the analysis of the 
steps, is appropriate to the investigation of the general truths concerning 
these subjects. 



1—2 



CHAPTER I 

PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS 

The notation adopted in the present work is based upon that of Peano, 
and the following explanations are to some extent modelled on those which 
he prefixes to his Formulario Mathematico. His use of dots as brackets is 
adopted, and so are many of his symbols. 

Variables. The idea of a variable, as it occurs in the present work, is 
more general than that which is explicitly used in ordinary mathematics. 
In ordinary mathematics, a variable generally stands for an undetermined 
number or quantity. In mathematical logic, any symbol whose meaning is not 
determinate is called a variable, and the various determinations of which its 
meaning is susceptible are called the values of the variable. The values may 
be any set of entities, propositions, functions, classes or relations, according 
to circumstances. If a statement is made about " Mr A and Mr B," " Mr A " 
and " Mr B " are variables whose values are confined to men. A variable may 
either have a conventionally-assigned range of values, or may (in the absence 
of any indication of the range of values) have as the range of its values all 
determinations which render the statement in which it occurs significant. 
Thus when a text-book of logic asserts that "A is A" without any indication 
as to what A may be, what is meant is that any statement of the form 
"A is A " is true. We may call a variable restricted when its values are 
confined to some only of those of which it is capable ; otherwise, we shall call 
it unrestricted. Thus when an unrestricted variable occurs, it represents any 
object such that the statement concerned can be made significantly {i.e. either 
truly or falsely) concerning that object. For the purposes of logic, the 
unrestricted variable is more convenient than the restricted variable, and we 
shall always employ it. We shall find that the unrestricted variable is still 
subject to limitations imposed by the manner of its occurrence, i.e. things 
which can be said significantly concerning a proposition cannot be said 
significantly concerning a class or a relation, and so on. But the limitations 
to which the unrestricted variable is subject do not need to be explicitly 
indicated, since they are the limits of significance of the statement in which 
the variable occurs, and are therefore intrinsically determined by this state- 
ment. This will be more fully explained later*. 

To sum up, the three salient facts connected with the use of the variable 

are: (1) that a variable is ambiguous in its denotation and accordingly undefined ; 

(2) that a variable preserves" a recognizable identity in various occurrences 

throughout the same context, so that many variables can occur together in the 

* Cf. Chapter II of the Introduction. 



CHAP. I] THE VARIABLE " 5 

same context each with its separate identity; and (3) that either the range of 
possible determinations of two variables may be the same, so that a possible 
determination of one variable is also a possible determination of the other, or 
the ranges of two variables may be different, so that, if a possible determina- 
tion of one variable is given to the other, the resulting complete phrase is 
meaningless instead of becoming a complete unambiguous proposition (true 
or false) as would be the case if all variables in it had been given any suitable 
determinations. 

The uses of various letters. Variables will be denoted by single letters, and 
so will certain constants ; but a letter which has once been assigned to a constant 
by a definition must not afterwards be used to denote a variable. The small 
letters of the ordinary alphabet will all be used for variables, except p and s 
after #40, in which constant meanings are assigned to these two letters. The 
following capital letters will receive constant meanings : B, C, D, E, F, /and J. 
Among small Greek letters, we shall give constant meanings to e, i and (at a 
later stage) to rj, 6 and &>. Certain Greek capitals will from time to time be 
introduced for constants, but Greek capitals will not be used for variables. Of 
the remaining letters, p, q, r will be called propositional letters, and will stand 
for variable propositions (except that, from #40 onwards, p must not be used 
for a variable); /, g, <f>, yfr, x, & an d (until #33) F will be called functional 
letters, and will be used for variable, functions. 

The small Greek letters not already mentioned will be used for variables 
whose values are classes, and will be referred to simply as Greek letters. Ordinary 
capital letters not already mentioned will be used for variables whose values 
are relations, and will be referred to simply as capital letters. Ordinary small 
letters other than p, q, r, s, f, g will be used for variables whose values are not 
known to be functions, classes, or relations; these letters will be referred to 
simply as small Latin letters. 

After the early part of the work, variable propositions and variable functions 
will hardly ever occur. We shall then have three main kinds of variables : 
variable classes, denoted by small Greek letters ; variable relations, denoted by 
capitals ; and variables not given as necessarily classes or relations, which will 
be denoted by small Latin letters. 

In addition to this usage of small Greek letters for variable classes, capital 
letters for variable relations, small Latin letters for variables of type wholly 
undetermined by the context (these arise from the possibility of "systematic 
ambiguity," explained later in the explanations of the theory of types), the 
reader need only remember that all letters represent variables, unless they have 
been defined as constants in some previous place in the book. In general the 
structure of the context determines the scope of the variables contained in it; 
but the special indication of the nature of the variables employed, as here 
proposed, saves considerable labour of thought. 



6 INTRODUCTION [CHAP. 

The fundamental functions of propositions. An aggregation of propositions, 
considered as wholes not necessarily unambiguously determined, into a single 
proposition more complex than its constituents, is a function with propositions 
as arguments. The general idea of such an aggregation of propositions, or of 
variables representing propositions, will not be employed in this work. But 
there are four special cases which are of fundamental importance, since all the 
aggregations of subordinate propositions into one complex proposition which 
occur in the sequel are formed out of them step by step. 

They are (1) the Contradictory Function, (2) the Logical Sum, or Dis- 
junctive Function, (3) the Logical Product, or Conjunctive Function, (4) the 
Implicative Function. These functions in the sense in which they are required 
in this work are not all independent ; and if two of them are taken as primitive 
undefined ideas, the other two can be defined in terms of them. It is to some 
extent — though not entirely — arbitrary as to which functions are taken as 
primitive. Simplicity of primitive ideas and symmetry of treatment seem to 
be gained by taking the first two functions as primitive ideas. 

The Contradictory Function with argument p, where p is any proposition, 
is the proposition which is the contradictory of p, that is, the proposition 
asserting that p is not true. This is denoted by ^p. Thus ^p is the 
contradictory function with p as argument and means the negation of the 
proposition p. It will also be referred to as the proposition not-p. Thus ~ p 
means not-p, which means the negation of p. 

The Logical Sum is a propositional function with two arguments p and q, 
and is the proposition asserting p or q disjunctively, that is, asserting that at 
least one of the two p and q is true. This is denoted by p v q. Thus p v q is 
the logical sum with p and q as arguments. It is also called the logical sum of 
p and q. Accordingly p v q means that at least p or q is true, not excluding the 
case in which both are true. 

The Logical Product is a propositional function with two arguments p and 
q, and is the proposition asserting p and q conjunctively, that is, asserting that 
both p and q are true. This is denoted by p . q, or — in order to make the dots 
act as brackets in a way to be explained immediately — by p : q, or by p :. q, 
or by p :: q. Thus p . q is the logical product, with p and q as arguments. It 
is also called the logical product of p and q. Accordingly p . q means that both 
p and q are true. It is easily seen that this function can be defined in terms 
of the two preceding functions. For when p and q are both true it must be 
false that either ~ p or ~ q is true. Hence in this book p . q is merely a 
shortened form of symbolism for 

~(~pv ~ q). 

If any further idea attaches to the proposition " both p and q are true," it is 
not required here. 



I] FUNCTIONS OF PROPOSITIONS 7 

The Implicative Function is a pro-positional function with two arguments 
p and q, and is the proposition that either not-p or q is true, that is, it is the 
proposition ~pv q. Thus if p is true, ~p is false, and accordingly the only 
alternative left by the proposition ~ p y q is that q is true. In other words 
if p and ~pvq are both true, then q is true. In this sense the proposition ;; 
~ pvq will be quoted as stating that p implies q. The idea contained in 
this propositional function is so important that it requires a symbolism which 
with direct simplicity represents the proposition as connecting p and q 
without the intervention of ~ p. But "implies" as used here expresses 
nothing eke than the connection between p and q also expressed by the 
disjunction "not-^ or q" The symbol employed for "p implies q" i.e. for 
" ~ pyq" is "pD q." This symbol may also be read "if p, then q." The 
association of implication with the use of an apparent variable produces 
an extension called " formal implication." This is explained later : it is an 
idea derivative from "implication" as here defined. When it is necessary 
explicitly to discriminate " implication " from " formal implication," it is called 
"material implication." Thus "material implication" is simply "implication" 
as here defined. The process of inference, which in common usage is often 
confused with implication, is explained immediately. 

These four functions of propositions are the fundamental constant {i.e. 
definite) propositional functions with propositions as arguments, and all other 
constant propositional functions with propositions as arguments, so far as they 
are required in the present work, are formed out of the.m by successive steps. 
No variable propositional functions of this kind occur in this work. 

Equivalence. The simplest example of the formation of a more complex 
function of propositions by the use of these four fundamental forms is furnished 
by "equivalence." Two propositions p and q are said to be "equivalent" 
when p implies q and q implies p. This relation between p and q is denoted 
by "p = q." Thus "p = q" stands for "(pDq).(q Op)." It is easily seen that 
two propositions are equivalent when, and only when, they are both true or 
are both false. Equivalence rises in the scale of importance when we come 
to "formal implication" and thus to "formal equivalence." It must not 
be supposed that two propositions which are equivalent are in any sense 
identical or even remotely concerned with the same topic. Thus " Newton 
was a man " and " the sun is hot " are equivalent as being both true, and 
" Newton was not a man " and " the sun is cold " are equivalent as being both 
false. But here we have anticipated deductions which follow later from our 
formal reasoning. Equivalence in its origin is merely mutual implication as 
stated above. 

Truth-values. The " truth-value " of a proposition is truth if it is true, 
and falsehood if it is false *. It will be. observed that the truth-values of 

* This phrase is due to Frege. 



8 INTRODUCTION [CHAP, 

pvq,p .q,pOq, ~p, p = q depend only upon those of p and q, namely the 
truth-value of'pvq" is truth if the truth-value of either p or q is truth, 
and is falsehood otherwise ; that of " p . q " is truth if that of both p and q is 
truth, and is falsehood otherwise ; that of "pDq" is truth if either that of p 
is falsehood or that of q is truth ; that of " ~ p " is the opposite of that of p ; 
and that of " p = q" is truth if p and q have the same truth-value, and is 
falsehood otherwise. Now the only ways in which propositions will occur 
in the present work are ways derived from the above by combinations and 
repetitions. Hence it is easy to see (though it cannot be formally proved 
except in each particular case) that if a proposition p occurs in any propo- 
sition f(p) which we shall ever have occasion to deal with, the truth-value 
°f f(p) w iH depend, not upon the particular proposition p, but only upon 
its truth-value ; i.e. if p = q, we shall have f(p) =f(q). Thus whenever two 
propositions are known to be equivalent, either may be substituted for the 
other in any formula with which we shall have occasion to deal. 

We' may call a function f(p) a " truth-function " when its argument p is 
a proposition, and the truth-value of f(p) depends only upon the truth- 
value of p. Such functions are by no means the only common functions of 
propositions. For example, "A believes p " is a function of p which will 
vary its truth-value for different arguments having the same truth-value : 
A may believe one true proposition without believing another, and may 
believe one false proposition without believing another. Such functions 
are not excluded from our consideration, and are included in the scope of 
any general propositions we may make about functions ; but the particular 
functions of propositions which we shall have occasion to construct or to con- 
sider explicitly are all truth-functions. This fact is closely connected with a 
characteristic of mathematics, namely, that mathematics is always concerned 
with extensions rather than intensions. The connection, if not now obvious, will 
become more so when we have considered the theory of classes and relations. 

Assertion-sign. The sign "b," called the "assertion-sign," means that 
what follows is asserted. It is required for distinguishing a complete propo- 
sition, which we assert, from any subordinate propositions contained in it but 
not asserted. In ordinary written language a sentence contained between full 
stops denotes an asserted proposition, and if it is false the book is in error. 
The sign "h" prefixed to a proposition serves this same purpose in our sym- 
bolism. For example, if " I- (p D p) " occurs, it is to be taken as a complete 
assertion convicting the authors of error unless the proposition "pOp" is 
true (as it is). Also a proposition stated in symbols without this sign " h " 
prefixed is not asserted, and is merely put forward for consideration, or as a 
subordinate part of an asserted proposition. 

Inference. The process of inference is as follows: a proposition "p" is 
asserted, and a proposition "p implies q" is asserted, and then as a sequel 



I] ASSERTION AND INFERENCE 9 

the proposition "q" is asserted. The trust in inference is the belief that if the 
two former assertions are not in error, the final assertion is not in error. 
Accordingly whenever, in symbols, where p and q have of course special 
determinations, 

"Vp" and "\-(pDq)" 
have occurred, then " h q " will occur if it is desired to put it on record. The 
process of the inference cannot be reduced to symbols. Its sole record is the 
occurrence of " h q." It is of course convenient, even at the risk of repetition, 
to write "\-p" and "b(pDq)" in close juxtaposition before proceeding to 
" h q" as the result of an inference. When this is to be done, for the sake of 
drawing attention to the inference which is being made, we shall write 
instead 

"hpDbq," 

which is to be considered as a mere abbreviation of the threefold statement 

" h p " and " h (p D q) " and " h q." 
Thus " bpDbq" may be read " p, therefore q," being in fact the same 
abbreviation, essentially, as this is; for " p, therefore q" does not explicitly 
state, what is part of its meaning, that p implies q. An inference is the 
dropping of a true premiss ; it is the dissolution of an implication. 

The use of dots. Dots on the line of the symbols have two uses, one to 
bracket off propositions, the other to indicate the logical product of two 
propositions. Dots immediately preceded or followed by " v " or " D " or 
" = " or " h," or by "(a)," «(*, y)," "(*, y, z)" . . . or "(a*)," " (a*, yV "(a*, 2/> *)'*• • • 
or "[(ix)(<f)x)]" or "[R'y]" or analogous expressions, serve to bracket off a 
proposition ; dots occurring otherwise serve to mark a logical product. The 
general principle is that a larger number of dots indicates an outside bracket, 
a smaller number indicates an inside bracket. The exact rule as to the scope 
of the bracket indicated by dots is arrived at by dividing the occurrences of 
dots into three groups which we will name I, II, and III. Group I consists of 
dots adjoining a sign of implication (D) or of equivalence (=) or of disjunction 
(v) or of equality by definition (= Df). Group II consists of dots following 
brackets indicative of an apparent variable, such as (w) or (x, y) or i^x) or 
{$%,y) or [(?#) (<£#)] or analogous expressions* Group III consists of dots 
which stand between propositions in order to indicate a logical product. 
Group I is of greater force than Group II, and Group II than Group III. 
The scope of the bracket indicated by any collection of dots extends backwards 
or forwards beyond any smaller number of dots, or any equal number from a 
group of less force, until we reach either the end of the asserted proposition 
or a greater number of dots or an equal number belonging to a group of 
equal or superior force. Dots indicating a logical product have a scope which 
works both backwards and forwards; other dots only work away from the 

* The meaning of these expressions will be explained later, and examples of the use of dots in 
connection with them will be given on pp. 16, 17. 



10 INTRODUCTION [CHAP. 

adjacent sign of disjunction, implication, or equivalence, or forward from the 
adjacent symbol of one of the other kinds enumerated in Group II. 

Some examples will serve to illustrate the use of dots. 

"pv q.D .qvp" means the proposition " 'p or q implies 'q or p.' " When 
we assert this proposition, instead of merely considering it, we write 

" h :pv q.D .qv p," 
where the two dots after the assertion-sign show that what is asserted is the 
whole of what follows the assertion-sign, since there are not as many as two 
dots anywhere else. If we had written "p : v : q . D . q v p" that would mean 
the proposition " either p is true, or q implies 'q or p.'" If we wished to assert 
this, we should have to put three dots after the assertion- sign. If we had 
written "pvq . D .q:v :p," that would mean the proposition " either 'p or q' 
implies q, orp is true." The forms "p . v .q . D .qv p" and "pv q . D .q . v .p" 
have no meaning. 

"p"Dq.D: qDr . D .pDr" will mean " if p implies q, then if q implies r, 
p implies r." If we wish to assert this (which is true) we write 
" H :. p D q . D : q D r . D . p D r." 

Again " pDq ,D .qDr :D .])Dr" will mean " if 'p implies q' implies 'q 
implies r,' then p implies r." This is in general untrue. (Observe that 
"P ^<l" ^ s sometimes most conveniently read as " p implies q," and sometimes 
as "if p, then q.") "pDq .qDr .D .p^>r" will mean "if p implies q, and 
q implies r, then p implies r." In this formula, the first dot indicates a logical 
product; hence the scope of the second dot extends backwards to the begin- 
ning df the proposition, "p D q : q D r . D . p D r" will mean "p implies q ; and 
if q implies r, then p implies r." (This is not true in general.) Here the two 
dots indicate a logical product ; since two dots do not occur anywhere else, the 
scope of these two dots extends backwards to the beginning of the proposition, 
and forwards to the end. 

"pvq .0 '..p • v . q Dr : D .pvr" will mean "if either p or q is true, then 
if either p or 'q implies r ' is true, it follows that either p or r is true." If 
this is to be asserted, we must put four dots after the assertion-sign, thus : 

"\- :: p v q . '3 :. p . v . q D r : D .pvr." 
(This proposition is proved in the body of the work ; it is *275.) If we wish 
to assert (what is equivalent to the above) the proposition: "if either p or q 
is true, and either p or 'q implies r' is true, then either p or r is true," we 

write 

" h :. p v q : p . v . qD r : D . p v r." 

Here the first pair of dots indicates a logical product, while the second pair 
does not. Thus the scope of the second pair of dots passes over the first pair, 
and back until we reacn the three dots after the assertion-sign. 

Other uses of dots follow the same principles, and will be explained as 
they are introduced. In reading a proposition, the dots should be noticed 



I] DEFINITIONS 11 

first, as they show its structure. In a proposition containing several signs of 
implication or equivalence, the one with the greatest number of dots before 
or after it is the principal one : everything that goes before this one is stated 
by the proposition to imply or be equivalent to everything that comes after it. 

Definitions. A definition is a declaration that a certain newly-introduced 
symbol or combination of symbols is to mean the same as a certain other 
combination of symbols of which the meaning is already known. Or, if the 
defining combination of symbols is one which only acquires meaning when 
combined in a suitable manner with other symbols*, what is meant is that 
any combination of symbols in which the newly-defined symbol or combination 
of symbols occurs is to have that meaning (if any) which results from substi- 
tuting the defining combination of symbols for the newly-defined symbol or 
combination of symbols wherever the latter occurs. We will give the names 
of definiendum and definiens respectively to what is defined and to that which 
it is defined as meaning. We express a definition by putting the definiendum 
to the left and the definiens to the right, with the sign "=" between, and the 
letters "Df " to the right of the definiens. It is to be understood that the 
sign " = " and the letters "Df " are to be regarded as together forming one 
symbol. The sign " =" without the letters "Df " will have a different meaning, 
to be explained shortly. 

An example of a definition is 

p"Dq. = . <^>p v q Df. 

It is to be observed that a definition is, strictly speaking, no part of the 
subject in which it occurs. For a definition is concerned wholly with the 
symbols, not with what they symbolise. Moreover it is not true or false, 
being the expression of a volition, not of a proposition. (For this reason, 
definitions are not preceded by the assertion-sign.) Theoretically, it is 
unnecessary ever to give a definition: we might always use the definiens 
instead, and thus wholly dispense with the definiendum. Thus although we 
employ definitions and do not define "definition," yet "definition" does not 
appear among our primitive ideas, because the definitions are no part of our 
subject, but are, strictly speaking, mere typographical conveniences. Prac- 
tically, of course, if we introduced no definitions, our formulae would very soon 
become so lengthy as to be unmanageable; but theoretically, all definitions are 
superfluous. 

In spite of the fact that definitions are theoretically superfluous, it is 
nevertheless true that they often convey more important information than is 
contained in the propositions in which they are used. This arises from two 
causes. First, a definition usually implies that the definiens is worthy of 
careful consideration. Hence the collection of definitions embodies our choice 

* This case will be fully considered in Chapter III of the Introduction. It need not further 
concern us at present. 



12 INTRODUCTION [CHAP. 

of subjects and our judgment as to what is most important. Secondly, when 
what is defined is (as often occurs) something already familiar, such as cardinal 
or ordinal numbers, the definition contains an analysis of a common idea, and 
may therefore express a notable advance. Cantor's definition of the continuum 
illustrates this: his definition amounts to the statement that what he is de- 
fining is the object which has the properties commonly associated with the 
word " continuum," though what precisely constitutes these properties had 
not before been known. In such cases, a definition is a " making definite ": it 
gives definiteness to an idea which had previously been more or less vague. 

For these reasons, it will be found, in what follows, that the definitions 
are what is most important, and what most deserves the reader's prolonged 
attention. 

Some important remarks must be made respecting the variables occurring 
in the dejiniens and the definiendum. But these will be deferred till the 
notion of an "apparent variable" has been introduced, when the subject can be 
considered as a whole. 

Summary of preceding statements. There are, in the above, three primi- 
tive ideas which are not " defined " but only descriptively explained. Their 
primitiveness is only relative to our exposition of logical connection and is 
not absolute; though of course such an exposition gains in importance ac- 
cording to the simplicity of its primitive ideas. These ideas are symbolised 
by "~jd" and "p v q," and by "h" prefixed to a proposition. 

Three definitions have been introduced: 

p 1 g. = ,oj(~pv~g') Df, 
p"Dq . = .~pv q Df, 

p = q . = ,p"D q.qDp Df. 

Primitive propositions. Some propositions must be assumed without proof, 
since all inference proceeds from propositions previously asserted. These, as 
far as they concern the functions of propositions mentioned above, will be 
found stated in #1, where the formal and continuous exposition of the subject 
commences. Such propositions will be called "primitive propositions." These, 
like the primitive ideas, are to some extent a matter of arbitrary choice; though, 
as in the previous case, a logical system grows in importance according as the 
primitive propositions are few and simple. It will be found that owing to the 
weakness of the imagination in dealing with simple abstract ideas no very 
great stress can be laid upon their obviousness. They are obvious to the in- 
structed mind, but then so are many propositions which cannot be quite true, 
as being disproved by their contradictory consequences. The proof of a logical 
system is its adequacy and its coherence. That is: (1) the system must embrace 
among its deductions all those propositions which we believe to be true and 
capable of deduction from logical premisses alone, though possibly they may 



I] PRIMITIVE PROPOSITIONS 13 

require some -slight limitation in the form of an increased stringency of enun- 
ciation; and (2) the system must lead to do contradictions, namely in pursuing, 
our inferences we must never be led to assert both p and not-p, i.e. both " I- .p" 
and "h ,~p" cannot legitimately appear. 

The following are the primitive propositions employed in the calculus of 
propositions. The letters "Pp" stand for "primitive proposition." 

(1) Anything implied by a true premiss is true Pp.. 
This is the rule which j ustifies inference. 

(2) Y'.pvp.S.p Pp, 

i.e. if p or p is true, then p is true. 

(3) bzq.D.pvq Pp, 

i.e. if q is true, then p or q is true. 

(4) h :p vq. D . q vp Pp, 

i.e. if p or q is true, then q or p is true. 

(5) h :pv(qv?~). D.qv(pvr) Pp, 

i.e. if either p is true or ,: q or r" is true, then either q is true or "p or r" is 
true. 

(0) I- :. qDr . D \p~vq. D .^ vr Pp, 
i.e. if </■ implies r, then "_p or q" implies "p or r." 

(7) Besides the above primitive propositions, we require a primitive pro- 
position called "the axiom of identification of real variables." When we have 
separately asserted two different functions of x, where x is undetermined, it 
is often important to know whether we can identify the x in one assertion 
with the x in the other. This will be the case — so our axiom allows us to 
infer — if both assertions present x as the argument to some one function, that 
is to say, if §x is a constituent in both assertions (whatever propositional func- 
tion may be), or, more generally, if (f>(x, y, z, ...) is a constituent in one 
assertion, and <f> (x, u, v, . . .) is a constituent in the other. This axiom introduces 
notions which have not yet been explained; for a fuller account, see the remarks 
accompanying *303, *1'7, *171, and *1*72 (which is the statement of this 
axiom) in the body of the work, as well as the explanation of propositional 
functions and ambiguous assertion to be given shortly. 

Some simple p?-opositions. In addition to the primitive propositions we 
have already mentioned, the following are among the most important of the 
elementary properties of propositions appearing among the deductions. 

The law of excluded middle: 

h .pv<^p. 

This is *211 below. We shall indicate in brackets the numbers given to the 
following propositions in the body of the work. 

The law of contradiction (*3 - 24): 



14 INTRODUCTION [CHAP. 

The law of double negation (#41 3) : 

The principle of transposition, i.e. "Up implies q, then not-q implies not-p," 
and vice versa: this principle has various forms, namely 
(#4-1) I- :pDq. = .~qD~p, 
(#4-11) t- :p = q. = f ~p = ~q, 
(#414) h i.p.q.D .r: = :p.^r.D .^q, 
as well as others which are variants of these. 
The law of tautology, in the two forms : 

(#4'24) H : p . = . p .p, 
(#425) Y:p. = .pyp, 
i.e. "p is true" is equivalent to "p is true and p is true," as well as to "p is true 
or p is true." From a formal point of view, it is through the law of tautology 
and its consequences that the algebra of logic is chiefly distinguished from 
ordinary algebra. 

The law of absorption: 

(#4-71) \-:.pDq.= :p. = .p.q, 
i.e. "p implies q" is equivalent to "p is equivalent to p . q." This is called the 
law of absorption because it shows that the factor q in the product is absorbed 
by the factor p, if p implies q. This principle enables us to replace an impli- 
cation (p D q) by an equivalence (p. = .p.q) whenever it is convenient to 
do so. 

An analogous and very important principle is the following: 
(*473) h:.q.D:p. = .p.q. 

Logical addition and multiplication of propositions obey the associative 
and commutative laws, and the distributive law in two forms, namely 
(#44) \- :. p . qv r . = : p . q . v . p . r, 
(#441) b :.p .v .q .r: = :pv q .pvr. 
The second of these distinguishes the relations of logical addition and multi- 
plication from those of arithmetical addition and multiplication. 

Propositioned functions. Let <frx be a statement containing a variable x 
and such that it becomes a proposition when x is given any fixed determined 
meaning. Then <f>x is called a "propositional function"; it is not a proposition, 
since owing to the ambiguity of x it really makes no assertion at all. Thus 
"x is hurt" really makes no assertion at all, till we have settled who x is. Yet 
owing to the individuality retained by the ambiguous variable x, it is an am- 
biguous example from the collection of propositions arrived at by giving all 
possible determinations to x in "x is hurt" which yield a proposition, true or 
false. Also if "x is hurt" and "y is hurt" occur in the same context, where y is 



I] PROPOSITIONAL FUNCTIONS 15 

another variable^ then according to the determinations given to x and y, they 
can be settled to be (possibly) the same proposition or (possibly) different 
propositions. But apart from some determination given to x and y, they retain 
in that context their ambiguous differentiation. Thus "x is hurt" is an am- 
biguous "value" of a propositional function. When we wish to speak of the 
propositional function corresponding to "x is hurt," we shall write "& is hurt." 
Thus "a> is hurt" is the propositional function and "x is hurt" is an ambiguous 
value of that function. Accordingly though "x is hurt" and "y is hurt" occurring 
in the same context can be distinguished, "ft is hurt" and "y is hurt" convey 
no distinction of meaning at all. More generally, <f>x is an ambiguous value of 
the propositional function <}>$, and when a definite signification a is substituted 
for x, <f>a is an unambiguous value of <f)x. 

Propositional functions are the fundamental kind from which the more usual 
kinds of function, such as "sin a;" or "logo?" or "the father of x," are derived. 
These derivative functions are considered later, and are called "descriptive 
functions." The functions of propositions considered above are a particular 
case of propositional functions. 

The range of values and total variation. Thus corresponding to any propo- 
sitional function <£a>, there is a range, or collection, of values, consisting of all 
the propositions (true or false) which can be obtained by giving every possible 
determination to x in <px. A value of x for which <f>x is true will be said to 
"satisfy" <]>£. Now in respect to the truth or falsehood of propositions of this 
range three important cases must be noted and symbolised. These cases are 
given by three propositions of which one at least must be true. Either (1) all 
propositions of the range are true, or (2) some propositions of the range are 
true, or (3) no proposition of the range is true. The statement (1) is symbolised 
by "(a?) . <f>x," and (2) is symbolised by "(g#) . <f>x." No definition is given of 
these two symbols, which accordingly embody two new primitive ideas in our 
system. The symbol "(x) . <j>x" may be read "<f>x always," or "<f>x is always true," 
or "<f>x is true for all possible values of x." The symbol "(ga?) . <f>x" may be 
read "there exists an x for which <j>x is true," or "there exists an x satisfying 
<f>&," and thus conforms to the natural form of the expression of thought. 

Proposition (3) can be expressed in terms of the fundamental ideas now on 
hand. In order to do this, note that " ~ <j>x" stands for the contradictory of <j>x. 
Accordingly ~ 4>cb is another propositional function such that each value of <f>8l 
contradicts a value of ~ <£&, and vice versa. Hence "(x) .^<f>x" symbolises the 
proposition that every value of <f>% is untrue. This is number (3) as stated above. 

It is an obvious error, though one easy to commit, to assume that cases 

(1) and (3) are each other's contradictories. The symbolism exposes this fallacy 

at once, for (1) is (x).<j>x, and (3) is (x).^<f>x, while the contradictory of (1) is 

oj {(x) . <f)x\. For the sake of brevity of symbolism a definition is made, namely 

~ (x) . §x . = . ~ [(x) . (f>x\ Df. 



16 INTRODUCTION [CHAP. 

Definitions of which the object is to gain some trivial advantage in brevity 
by a slight adjustment of symbols will be said to be of "merely symbolic import," 
in contradistinction to those definitions which invite consideration of an im- 
portant idea. 

The proposition (x) . <f>x is called the "total variation" of the function <££. 

For reasons which will be explained in Chapter II, we do not take negation 
as a primitive idea when propositions of the forms (x) . <f>x and fax) . <f>x are 
concerned, but we define the negation of (x) . <f>x, i.e. of "<f>x is always true," as 
being "<f>x is sometimes false," i.e. "fax) . ~<$>x" and similarly we define the 
negation of fax) . 4>x as being (x) . ~<f>x. Thus we put 
~ {(x) . <t>x} . = . fax) . ~ <f>x Df, 

~ {(3^) • $*'} ■ = ■(#)■ ~ <f>® 1^ 

In like manner we define a disjunction in which one of the propositions is 
of the form "(x) . <j>x" or "fax) . <f>x" in terms of a disjunction of propositions 
not of this form, putting 

(x) . <px . v . p : = . (x) . <fix v p Df, 
i.e. "either <f>x is always true, orp is true" is to mean "'fyxoxp' is always true," 
with similar definitions in other cases. This subject is resumed in Chapter II, 
and in #9 in the body of the work. 

Apparent variables. The symbol "(x) . <f>x" denotes one definite proposition, 
and there is no distinction in meaning between "(x) . <px" and "(y) . fyy" when 
they occur in the same context. Thus the "a?" in "(x) . <j>x" is not an ambiguous 
constituent of any expression in which "(.:). <f>x" occurs; and such an ex- 
pression does not cease to convey a determinate meaning by reason of the 
ambiguity of the x in the "<f>x." The symbol "(x) . (f>x" has some analogy to 
the symbol 

<j> (x) dx 



for definite integration, since in neither case is the expression a function of x. 

The range of x in "(x).<f>x" or "fax).<f>x" extends over the complete 
field of the values of x for which "<f>x" has meaning, and accordingly the 
meanino- of "(x) . <f>x" or "fax) . <f>x" involves the supposition that such a field 
is determinate. The x which occurs in "(#).</>#" or "fax).<f>x" is called 
(following Peano) an " apparent variable." It follows from the meaning of 
"fax).<f>x" that the x in this expression is also an apparent variable. A 
proposition in which x occurs as an apparent variable is not a function of x. 
Thus e.g. "(.r) .x = x" will mean "everything is equal to itself." This is an 
absolute Constant, not a function of a variable x. This is why the x is called 
an apparent variable in such cases. 

Besides the "range" of x in "(x).<f>x" or "fax).<f>x," which is the field 
of the values that x may have, we shall speak of the "scope" of x, meaning 



I] APPARENT VARIABLES 17 

the function of which all values or some value are being affirmed. If we are 
asserting all values (or some value) of"<f>x," "<f>x" is the scope of #; if we are 
asserting all values (or some value) of "<j>xDp," "<j>xDp" is the scope of x; 
if we are asserting all values (or some value) of "<f>x D yfrx" "<f>x D yfrx" will be 
the scope of x, and so on. The scope of x is indicated by the number of dots 
after the "(#)" or "(g#)"; that is to say, the scope extends forwards until 
we reach an equal number of dots not indicating a logical product, or a greater 
number indicating a logical product, or the end of the asserted proposition in 
which the "(#)" or "(3#)" occurs, whichever of these happens first*. Thus e.g. 

"(x) :<f>x.D. yjrx" 
will mean "<j>x always implies -tyx," but 

"(x).(f>x.0.ylrx" 
will mean "if <f>x is always true, then yjrx is true for the argument x." 
Note that in the proposition 

(x) . <f>x . D . yp-x 
the two x'a have no connection with each other. Since only one dot follows 
the x in brackets, the scope of the first x is limited to the "(j>x" immediately 
following the x in brackets: It usually conduces to clearness to write 

(x) . <f>x . D . yjry 
rather than (x) . <f>x . D . yfrx, 

since the use of different letters emphasises the absence of connection between 
the two variables; but there is no logical necessity to use different letters, 
and it is sometimes convenient to use the same letter. 

Ambiguous assertion and the real variable. Any value "<fix" of the function 
$x can be asserted. Such an assertion of an ambiguous member of the values 
of $ob is symbolised by 

"h.^x." 

Ambiguous assertion of this kind is a primitive idea, which cannot be defined 
in terms of the assertion of propositions. This primitive idea is the one which 
embodies the use of the variable. Apart from ambiguous assertion, the con- 
sideration of "<f>x," which is an ambiguous member of the values of cf>x, would 
be of little consequence. When we are considering or asserting "<f>x," the 
variable x is called a " real variable." Take, for example, the law of excluded 
middle in the form which it has in traditional formal logic : 

" a is either b or not b." 
Here a and b are real variables: as they vary, different propositions are 
expressed, though all of them are true. While a and b are undetermined, as in 
the above enunciation, no one definite proposition is asserted, but what is 
asserted is any value of the propositional function in question. This can only 

* This agrees with the rules for the occurrences of dots of the type of Group II as explained 
above, pp. 9 and 10. 

R&W I 2 



18 INTRODUCTION [CHAP; 

be legitimately asserted if, whatever value may be chosen, that value is true, 

i.e. if all the values are true. Thus the above form of the law of excluded 

middle is equivalent to 

" (a, b) . a is either b or not b," 

i.e. to "it is always true that a is either b or not b." But these two, though 
equivalent, are not identical, and we shall find it necessary to keep them 
distinguished. 

When we assert something containing a real variable, as in e.g. 

"K# = #," 
we are asserting any value of a prepositional function. When we assert some- 
thing containing an apparent variable, as in 

" h . (x) . x = x" 
or "V ,{^x).x = x" 

we are asserting, in the first case all values, in the second case some value 
(undetermined), of the propositional function in question. It is plain that 
we can only legitimately assert " any value " if all values are true; for other- 
wise, since the value of the variable remains to be determined, it might be so 
determined as to give a false proposition. Thus in the above instance, since 

we have 

\- ,x = x 

we may infer h.(x).x — x. 

And generally, given an assertion containing a real variable x, we may trans- 
form the real variable into an apparent one by placing the x in brackets at 
the beginning, followed by as many dots as there are after the assertion-sign. 
When we assert something containing a real variable, we cannot strictly 
be said to be asserting a proposition, for we only obtain a definite proposition 
by assigning a value to the variable, and then our assertion only applies to 
one definite case, so that it has not at all the same force as before. When what 
we assert contains a real variable, we are asserting a wholly undetermined one 
of all the propositions that result from giving various values to the variable. 
It will be convenient to speak of such assertions as asserting a propositional 
function. The ordinary formulae of mathematics contain such assertions; for 

example 

"sin 2 # + cos 2 # = 1" 

does not assert this or that particular case of the formula, nor does it assert 
that the formula holds for all possible values of x, though it is equivalent to 
this latter assertion; it simply asserts that the formula holds, leaving x wholly 
undetermined; and it is able to do this legitimately, because, however x may 
be determined, a true proposition results. 

Although an assertion containing a real variable does not, in strictness, 
assert a proposition, yet it will be spoken of as asserting a proposition except 
when the nature of the ambiguous assertion involved is under discussion. 



I] SEAL VARIABLES 19 

Definition and real variables. When the definiens contains one or more 
real variables, the definiendum must also contain them. For in this case we 
have a function of the real variables, and the definiendum must have the same 
meaning as the definiens for all values of these variables, which requires that 
the symbol which is the definiendum should contain the letters representing 
the real variables. This rule is not always observed by mathematicians, and 
its infringement has sometimes caused important confusions of thought, 
notably in geometry and the philosophy of space. 

In the definitions given above of "p . q" and "p Dq" and "p = q," p and q 
are real variables, and therefore appear on both sides of the definition. In 
the definition of "~ {{%) . <f>x}" only the function considered, namely <f>z, is a 
real variable; thus so far as concerns the rule in question, x need not appear 
on the left. But when a real variable is a function, it is necessary to indicate 
how the argument is to be supplied, and therefore there are objections to 
omitting an apparent variable where (as in the case before us) this is the 
argument to the function which is the real variable. This appears more 
plainly if, instead of a general function <£&, we take some particular function, 
say "ob ~ a" and consider the definition of ~ {(x) . x = a). Our definition gives 

~ {(x) . x = a] . = . (a#) . ~ (x = a) Df. 
But if we had adopted a notation in which the ambiguous value "x = a" 
containing the apparent variable x, did not occur in the definiendum, we 
should have had to construct a notation employing the function itself, namely 
"& = a." This does not involve an apparent variable, but would be clumsy in 
practice. In fact we have found it convenient and possible — except in the 
explanatory portions — to keep the explicit use of symbols of the type "<£&," 
either as constants [e.g. £=a] or as real variables, almost entirely out of this 
work. 

Propositions connecting real and apparent variables. The most important 
propositions connecting real and apparent variables are the following: 

(1) " When a propositional function can be asserted, so can the proposition 
that all values of the function are true." More briefly, if less exactly, " what 
holds of any, however chosen, holds of all." This translates itself into the rule 
that when a real variable occurs in an assertion, we may turn it into an apparent 
variable by putting the letter representing it in brackets immediately after 
the assertion-sign. 

(2) " What holds of all, holds of any," i.e. 

h : (x) . <f>x . D . $y. 
This states " if <f>x is always true, then <\>y is true." 

(3) "If <f>y is true, then <f>x is sometimes true," i.e. 

\-:<f>y.D. (rx) . <t>x. 

2—2 



20 INTRODUCTION [CHAP. 

An asserted proposition of the form " (g«) . </>&' " expresses an " existence- 
theorem," namely " there exists an x for which <f>x is true." The above pro- 
position gives what is in practice the only way of proving existence-theorems: 
we always have to find some particular y for which <f>y holds, and thence to 
infer " (g#) . <f>x.". If we were to assume what is called the multiplicative 
axiom, or the equivalent axiom enunciated by Zermelo, that would, in an 
important class of cases, give an existence-theorem where no particular instance 
of its truth can be found. 

In virtue of " h : (x) ,<f)x.D.<f>y" and " h : $y . D . (g#) . <£#," we have 
" I- : (x) .<j>x.D. (g#) . (j>x," i.e. " what is always true is sometimes true." This 
would not be the case if nothing existed; thus our assumptions contain the 
assumption that there is something. This is involved in the principle that 
what holds of all, holds of any; for this would not be true if there were no 
"any." 

(4) " If <j>x is always true, and yfrx is always true, then ' <f>x . yjrw ' is always 
true," i.e. 

h : . (x) . (f>x : (x) . yfrx : D . (x) . <f>x . \^x. 

(This requires that <f> and yfr should be functions which take arguments of the 
same type. We shall explain this requirement at a later stage.) The converse 
also holds ; i.e. we have 

h : . (x) . (f>x . yfrx ."D:(x). <f>x : (x) . yjrx. 

It is to some extent optional which of the propositions connecting real 
and -apparent variables are taken as primitive propositions. The primitive 
propositions assumed, on this subject, in the body of the work (*9), are the 
following : 

(1) I" : <f>x . D . ( a *) . <\>z. 

(2) h:4>xv<f>y.D.(^z).cf>z, 

i.e. if either </>.*.■ is true, or 4>y is true, then (rz) . cj>z is true. (On the necessity 
for this primitive proposition, see remarks on #9-11 in the body of the work.) 

(3) If we can assert <f>y, where y is a real variable, then we can assert 
(x) . <px ; i.e. what holds of any, however chosen, holds of all. 

Formal implication and formal equivalence. When an implication, say 
<f>x . D . yfrx, is said to hold always, i.e. when (x) : <f>x . D . \jrx, we shall say that 
<f)x formally implies yfrx ; and propositions of the form " (x) z^.D.^x " will 
be said to state formal implications. In the usual instances of implication, 
such as "'Socrates is a man' implies 'Socrates is mortal,' "'we have a propo- 
sition of the form " 4>x . D . \]rx " in a case in which " (x) :</>*•. D . yfrx " is true. 
In such a case, we feel the implication as a particular case of a formal impli- 
cation. Thus it has come about that implications which are not particular 
cases of formal implications have not been regarded as implications at all. 
There is also a practical ground for the neglect of such implications, for, speaking 



I] FORMAL IMPLICATION 21 

generally, they can only be known when it is already known either that their 
hypothesis is false or that their conclusion is true ; and in neither of these 
cases do they serve to make us know, the conclusion, since in the first case the 
conclusion need not be true, and in the second it is known already. Thus 
such implications do not serve the purpose for which implications are chiefly 
useful, namely that of making us know, by deduction, conclusions of which we 
were previously ignorant. Formal implications, on the contrary, do serve this 
purpose, owing to the psychological fact that we often know "(x):^."^.^" 
and <f>y, in cases where -tyy (which follows from these premisses) cannot easily 
be known directly. 

These reasons, though they do not warrant the complete neglect of impli- 
cations that are not instances of formal implications, are reasons which make 
formal implication very important. A formal implication states that, for all 
possible values of x, if the hypothesis $x is true, the conclusion yjrx is true. 
Since " $x . D . yfrx " will always be true when <f>x is false, it is only the values 
of x that make <f>x true that are important in a formal implication ; what is 
effectively stated is that, for all these values, tyx is true. Thus propositions 
of the form " all a is /3," " no a is /9 " state formal implications, since the first 
(as appears by what has just been said) states 

(x) : x is an a . D . x is a /3, 

while the second states 

(x) : x is an a . D . x is not a yS. 

And any formal implication " (x) : </># . D . yjrx " may be interpreted as : " All 
values of x which satisfy* <f>x satisfy tyx," while the formal implication 
" (x):<f)X.D .royfrx " may be interpreted as : " No values of x which satisfy <f>x 
satisfy tyx." 

We have similarly for " some a is /3 " the formula 
(g#) . x is an a . x is a /3, 
and for " some a is not /3 " the formula 

(g#) . x is an a . x is not a /9. 

Two functions <f>x, yfrx are called formally equivalent when each always 
implies the other, i.e. when 

(x) : <f>x . = . yfrx, 

and a proposition of this form is called a formal equivalence. In virtue of 
what was said about truth-values, if <f>x and tyx are formally equivalent, either 
may replace the other in any truth-function. Hence for all the purposes of 
mathematics or of the present work, <pz may replace yjrz or vice versa in any 
proposition with which we shall be concerned. Now to say that <f>x and tyx 
are formally equivalent is the same thing as to say that </>2 and yjrz have the 
same extension, i.e. that any value of x which satisfies either satisfies the other. 
* A value of x is said to satisfy <f>x or tj>x when <f>x is true for that value of x. 



22 INTRODUCTION [CHAP. 

Thus whenever a constant function occurs in our work, the truth-value of the 
proposition in which it occurs depends only upon the extension of the function. 
A proposition containing a function </>£ and having this property (i.e. that its 
truth-value depends only upon the extension of <fiz) will be called an exten- 
sional function of (f>z . Thus the functions of functions with which we shall be 
specially concerned will all be extensional functions of functions. 

What has just been said explains the connection (noted above) between 
the fact that the functions of propositions with which mathematics is specially 
concerned are all truth-functions and the fact that mathematics is concerned 
with extensions rather than intensions. 

Convenient abbreviation. The following definitions give alternative and often 
more convenient notations : 

<f>x . D x . yjrw : = : (x) : <f)X . D . -tyx Df, 

<px . = x . ifrx : = :(x):<f>x. = . tyx Df. 

This notation " <f>x . O x . yfrx " is due to Peano, who, however, has no notation 

for the general idea " (x) . <J3X." It may be noticed as an exercise in the use 

of dots as brackets that we might have written 

<px D x ijrx . — . (x) . <f>% D ijrx Df, 
cf>x = x yfrx . = .(#). <f>x = yjrx . Df. 
In practice however, when </>£ and yjr^c are special functions, it is not possible 
to employ fewer dots than in the first form, and often more are required. 

The following definitions give abbreviated notations for functions of two 
or more variables : 

(x, y) . <f> (x, y) . = : (x) : (y) . <£ (x, y) Df, 
and so on for any number of variables ; 

</> (x, y) . D x>y .^(x,y): = : (x, y) : 4>(x,y) .1 .^ (x, y) Df, 
and so on for any number of variables. 

Identity. The propositional function " x is identical with y " is expressed by 

x = y. 
This will be defined (cf. *13-01), but, owing to certain difficult points involved 
in the definition, we shall here omit it (cf. Chapter II). We have, of course, 
\- .x = x (the law of identity), 
\-:x = y. = .y = x, 
\-\x = y.y = z."5.x = z. 
The first of these expresses the reflexive property of identity : a relation is 
called reflexive when it holds between a term and itself, either universally, or 
whenever it holds between that term and some term. The second of the 
above propositions expresses that identity is a symmetrical relation : a relation 
is called symmetrical if, whenever it holds between x and y, it also holds 



I] IDENTITY 23 

between y and a;. The third proposition expresses that identity is a transitive 
relation : a relation is called transitive if, whenever it holds between x and y 
and between y and z, it holds also between x and z. 

We shall find that no new definition of the sign of equality is required in 
mathematics : all mathematical equations in which the sign of equality is used 
in the ordinary way express some identity,, and thus use the sign of equality 
in the above sense. 

If x and y are identical, either can replace the other in any proposition 
without altering the truth- value of the proposition; thus we have 

r- : x ■= y . D . <f>x = <j>y. 
This is a fundamental property of identity, from which the remaining properties 
mostly follow. 

It might be thought that identity would not have much importance, since 
it can only hold between x and y if x and y are different symbols for the same 
object. This view, however, does not apply to what we shall call " descriptive 
phrases," i.e. " the so-and-so." It is in regard to such phrases that identity is 
important, as we shall shortly explain. A proposition such as " Scott was the 
author of Waverley " expresses an identity in which there is a descriptive 
phrase (namely " the author of Waverley ") ; this illustrates how, in such cases, 
the assertion of identity may be important. It is essentially the same case 
when the newspapers say " the identity of the criminal has not transpired." 
In such a case, the criminal is known by a descriptive phrase, namely " the 
man who did the deed," and we wish to find an x of whom it is true that 
" #=the man who did the deed." When such an x has been found, the identity 
of the criminal has transpired. 

Classes and relations. A. class (which is the same as a manifold or aggre- 
gate) is all the objects satisfying some propositional function. If a is the class 
composed of the objects satisfying <px, we shall say that a is the class determined 
by <£& Every propositional function thus determines a class, though if the 
propositional function is one which is always false, the class will be null, 
i.e. will have no members. The class determined by the function <f>% will be 
represented by z (<f>z)*. Thus for example if <f>x is an equation, z (<f>z) will be 
the class of its roots ; if <f>x is " x has two legs and no feathers," z (<j>z) will 
be the class of men ; if <f>x is " < x < 1," z (<f>z) will be the class of proper 
fractions, and so on. 

It is obvious that the same class of objects will have many determining 
functions. When it is not necessary to specify a determining function of a 
class, the class may be conveniently represented by a single Greek letter. 
Thus Greek letters, other than those to which some constant meaning is 
assigned, will be exclusively used for classes. 

* Any other letter may be used instead of z. 



24 INTRODUCTION [CHAP. 

There are two kinds of difficulties which arise in formal logic ; one kind 
arises in connection with classes and relations and the other in connection 
with descriptive functions. The point of the difficulty for classes and relations, 
so far as it concerns classes, is that a class cannot be an object suitable as an 
argument to any of its determining functions. If a represents a class and <j>x 
one of its determining functions [so that a = z {<f>z)\ it is not sufficient that 
<]>a be a false proposition, it must be nonsense. Thus a certain classification 
of what appear to be objects into things of essentially different types seems 
to be rendered necessary. This whole question is discussed in Chapter II, on 
the theory of types, and the formal treatment in the systematic exposition, 
which forms the main body of this work, is guided by this discussion. The 
part of the systematic exposition which is specially concerned with the theory 
of classes is #20, and in this Introduction it is discussed in Chapter III. It is 
sufficient to note here that, in the complete treatment of #20, we have avoided 
the decision as to whether a class of things has in any sense an existence as 
one object. A decision of this question in either way is indifferent to our logic, 
though perhaps, if we had regarded some solution which held classes and re- 
lations to be in some real sense objects as both true and likely to be universally 
received, we might have simplified one or two definitions and a few preliminary 
propositions. Our symbols, such as " ct (<f>%) " and a and others, which represent 
classes and relations, are merely defined in their use, just as V 2 , standing for 

dx* + df + dz* ' 
has no meaning apart from a suitable function of x, y, z on which to operate. 
The result of our definitions is that the way in which we use classes corre- 
sponds in general to their use in ordinary thought and speech ; and whatever 
may be the ultimate interpretation of the one is also the interpretation of 
the other. Thus in fact our classification of types in Chapter II really 
performs the single, though essential, service of justifying us in refraining 
from entering on trains of reasoning which lead to contradictory conclusions. 
The justification is that what seem to be propositions are really nonsense. 

The definitions which occur in the theory of classes, by which the idea of 
a class (at least in use) is based on the other ideas assumed as primitive, 
cannot be understood without a fuller discussion than can be given now 
(cf. Chapter II of this Introduction and also #20). Accordingly, in this pre- 
liminary survey, we proceed to state the more important simple propositions 
which result from those definitions, leaving the reader to employ in his mind 
the ordinary unanalysed idea of a class of things. Our symbols in their usage 
conform to the ordinary usage of this idea in language. It is to be noticed 
that in the systematic exposition our treatment of classes and relations requires 
no new primitive ideas and only two new primitive propositions, namely the 
two forms of the "Axiom of Reducibility " (cf. next Chapter) for one and two 
variables respectively. 



I] CLASSES 25 

The prepositional function "a? is a member of the class a" will be expressed, 
following Peano, by the notation 

area. 

Here e is chosen as the initial of the word earL " x e a " may be read " x is 
an a." Thus "x e man" will mean "x is a man," and so on. For typographical 
convenience we shall put 

x~ea. = .~(xea.) Df, 

x, y eOL. = .xea. .y ea Df. 

For " class " we shall write " Cls "; thus " ae Cls " means " a is a class." 

We have 

h : x e z (<f>z) . = • <f>x, 

i.e. "'x is a member of the class determined by <f>z' is equivalent to 'x 
satisfies <j)z,' or to ' <f>x is true.' " 

A class is wholly determinate when its membership is known, that is, there 

cannot be two different classes having the same membership. Thus if <f>x, yfrx 

are formally equivalent functions, they determine the same class ; for in that 

case, if a? is a member of the class determined by <££, and therefore satisfies <f>x, 

it also satisfies tyx, and is therefore a member of the class determined by yfrfc. 

Thus we have 

h :. z (<f)z) = z (tyz) . = m .(f)X.= x . yfrx. 

The following propositions are obvious and important : 

h :. a = 2 (<j>z) . = : x e a . = x . <f>x, 

i.e. a is identical with the class determined by <fiz when, and only when, "x is 
an a " is formally equivalent to (f>x; 

b :.a = ft ,= :x60t.= x .xe/3, 

i.e. two classes o and ft are identical when, and only when, they have the same 
membership ; 

h . & (x € a) = a, 

i.e. the class whose determining function is " x is an o " is a, in other words, 
a is the class of objects which are members of a ; 

r-.2(£*)eCls, 

i.e. the class determined by the function <pz is a class. 

It Will be seen that, according to the above, any function of one variable 
can be replaced by an equivalent function of the form "xea." Hence any 
extensional function of functions which holds when its argument is a function 
of the form "zea," whatever possible value a may have, will hold also when 
its argument is any function <f>z. Thus variation of classes can replace varia- 
tion of functions of one variable in all the propositions of the sort with which 
we are concerned. 



26 INTRODUCTION [CHAP. 

In an exactly analogous manner we introduce dual or dyadic relations, 
i.e. relations between two terms. Such relations will be called simply 
"relations"; relations between more than two terms will be distinguished as 
multiple relations, or (when the number of their terms is specified) as triple, 
quadruple, . . . relations, or as triadic, tetradic, . . . relations. Such relations will 
not concern us until we come to Geometry. For the present, the only relations 
we are concerned with are dual relations. 

Relations, like classes, are to be taken in extension, i.e. if R and S are 
relations which hold between the same pairs of terms, R and S are to be 
identical. We may regard a relation, in the sense in which it is required for 
our purposes, as a class of couples ; i.e. the couple (x, y) is to be one of the 
class of couples constituting the relation R if x has the relation R to y*. 
This view of relations as classes of couples will not, however, be introduced 
into our symbolic treatment, and is only mentioned in order to show that it 
is possible so to understand the meaning of the word relation that a relation 
shall be determined by its extension. 

Any function (f> (x, y) determines a relation R between x and y. If we 
regard a relation as a class of couples, the relation determined by <£ (x, y) is 
the class of couples (x, y) for which </> (x, y) is true. The relation determined 
by the function <£ (x, y) will be denoted by 

We" shall use a capital letter for a relation when it is not necessary to specify 
the determining function. Thus whenever a capital letter occurs, it is to be 
understood that it stands for a relation. 

The propositional function " x has the relation R to y " will be expressed 
by the notation 

xRy. 

This notation is designed to keep as near as possible to common language, 
which, when it has to express a relation, generally mentions it between its 
terms, as in " x loves y," " x equals y," " x is -greater than y," and so on. For 
" relation " we shall write " Rel "; thus " R e Rel " means "R is a relation." 

Owing to our taking relations in extension, we shall have 

r- :. x§<f> (x, y) = x§yfr (x,y).= :<f> (x, y) . = x , y .ty 0> y), 

i.e. two functions of two variables determine the same relation when, and only 
when, the two functions are formally equivalent. 

We have V . z {xf/rf* (x, y)}w . = ,<f> (z, w), 

* Such a couple has a sense, i.e. the couple (x, y) is different from the couple (y, x), unless 
x = y. We shall call it a "couple with sense," to distinguish it from the class consisting of x 
and y. It may also be called an ordered couple. 



I] CALCULUS OF CLASSES 27 

i.e. "z has to w the relation determined by the function <j> (x, y)" is equivalent 

to <f> (z, w) ; 

I- :. R = %<£ (x, y) . = : #% . = x ,y<f> («, 2/)> 

h :.R = S . = :xRy .= Xjy .xSy, 

\-.$f/(xRy) = R, 

K®^(a?,y)}eRel. 

These propositions are analogous to those previously given for classes. It 
results from them that any function of two variables is formally equivalent to 
some function of the form xRy; hence, in extensional functions of two variables, 
variation of relations can replace variation of functions of two variables. 

Both classes and relations have properties analogous to most of those of 
propositions that result from negation and the logical sum. The logical product 
of two classes a and /3 is their common part, i.e. the class of terms which are 
members of both. This is represented by a r\ fi. Thus we put 

ar\fi = <x}(xea.xe0) Df. 

This gives us I- : x e a r\ . = . x e a . x e {3, 

i.e. "x is a member of the logical product of a and 0" is equivalent to the 
logical product of " x is a member of a " and " x is a member of fi" 

Similarly the logical sum of two classes a and is the class of terms which 
are members of either ; we denote it by a u 0. The definition is 

ayj@ = x(x€a.v.xefi) Df, 
and the connection with the logical sum of propositions is given by 
V \.xeoL\J fi . = '.xea..v .xe ft. 

The negation of a class a consists of those terms x for which "xea" can 
be significantly and truly denied. We shall find that there are terms of other 
types for which "xea" is neither true nor false, but nonsense. These terms 
are not members of the negation of a. 

Thus the negation of a class a is the class of terms of suitable type which 
are not members of it, i.e. the class tc(x~ea). We call this class ",-a" (read 
"hot-a"); thus the definition is 

— a = x(x<^e a) Df, 
and the connection with the negation of propositions is given by 

h : xe — a. = . x<^>ea. 

In place of implication we have the relation of inclusion. A class a is said 

to be included or contained in a class if all members of a are members of 0, 

i.e. if x e a . D x . x e 0. We write " a C /3 " for " a is contained in 0." Thus we 

put 

aC/8. = :xea. D x .xeft Df. 



28 INTRODUCTION [CHAP. 

Most of the formulae concerning p . q, p v q, ~p, pD q remain true if we 
substitute a r\ ft, a u ft, — a, a C ft. In place of equivalence, we substitute 
identity ; for " p = q " was denned as "pDq.q Dp" but " a C ft . ft C a " gives 
"x e a . =3. . x e ft," whence a = ft. 

The following are some propositions concerning classes which are analogues 
of propositions previously given concerning propositions : 

h ? an/3 = -(-au-/3), 
i.e. the common part of a and ft is the negation of " not-a or not-/3 " ; 

V . x e (a u — a), 
i.e. " x is a member of a or not-a " ; 

h . #~e (a n — a), 
i.e. " # is not a member of both a and not-a " ; 

h.a = -(-a), 
h:aCft. = .-ftC-a, 
\-:a = ft. = .-a = -ft, 
h : a = a n a, 
h« = au«. 
The two last are the two forms of the law of tautology. 
The law of absorption holds in the form 

\- : aCft . = .a= ar\ ft. 

Thus for example " all Cretans are liars " is equivalent to " Cretans are 
identical with lying Cretans." 

Just as we have hzjj'Dq.qDr.D.p'Dr, 

so we have I- : a Cft . ftCy . D . aC^ 

This expresses the ordinary syllogism in Barbara (with the premisses 
interchanged) ; for " a C ft " means the same as " all a's are ft's," so that the 
above proposition states : '" If all a's are ft's, and all ft's are 7's, then all a's 
are 7's." (It should be observed that syllogisms are traditionally expressed 
with " therefore," as if they asserted both premisses and conclusion. This is, 
of course, merely a slipshod way of speaking, since what is really asserted is 
only the connection of premisses with conclusion.) 

The syllogism in Barbara when the minor premiss has an individual 
subject is 

h : x e ft . ft C 7 . D . x e 7, 

e.g. " if Socrates is a man, and all men are mortals, then Socrates is a 
mortal." This, as was pointed out by Peano, is not a particular case of 
"aC/3./3C7.D.aC 7," since " x e ft " is not a particular case of " a C ft." 
This point is important, since traditional logic is here mistaken. The nature 
and magnitude of its mistake will become clearer at a later stage. 



I] CALCULUS OP CLASSES 29 

For relations, we have precisely analogous definitions and propositions. 
We put 

knS=$p (xRy . xSy) Df, 

which leads to h : x ( R n S) y . = . xRy . xSy. 

Similarly RvS = $y (xRy . v . xSy) Df, 

^-R = xy {-{xRy)} Df, 

RGS. = : xRy . D^ . xSy Df. 

Generally, when we require analogous but different symbols for relations 
and for classes, we shall choose for relations the symbol obtained by adding 
a dot, in some convenient position, to the corresponding symbol for classes. 
(The dot must not be put on the line, since that would cause confusion with 
the use of dots as brackets.) But such symbols require and receive a special 
definition in each case. 

A class is said to exist when it has at least one member : " a exists " is 
denoted by " g ! a." Thus we put 

3 ! a . = . (g#) . x e a Df. 
The class which has no members is called the " null-class," and is denoted by 
"A." Any propositional function which is always false determines the null- 
class. One such function is known to us already, namely "x is not identical 
with x," which we denote by " x 4= x." Thus we may use this function for de- 
fining A, and put 

A = x(x$x) Df. 

The class determined by a function which is always true is called the 
universal class, and is represented by V; thus 

V = &(# = #) Df. 

Thus A is the negation of V. We have 

h . (x) .xeY, 
i.e. " ' x is a member of V ' is always true " ; and 

J- . (x) . x~e A, 
i.e. " 'x is a member of A' is always false." Also 

h : a = A . = . ~g ! a, 
i.e. " a is the null-class " is equivalent to " a does not exist." 

For relations we use similar notations. We put 
3 ! R . = . (g#, y) . xRy, 
i.e. " a ! R " means that there is at least one couple x, y between which 
the relation R holds. A will be the relation which never holds, and V the 
relation which always holds. V is practically never required ; A will be the 
relation xy (x ^ x . y 4= y). We have 

Y.{x,y).~{xky), 
and h : R = A. = . ~g! R. 



30 INTRODUCTION [CHAP. 

There are no classes which contain objects of more than one type. Ac- 
cordingly there is a universal class and a null-class proper to each type of 
object. But these symbols need not be distinguished, since it will be found 
that there is no possibility of confusion. Similar remarks apply to relations. 

Descriptions. By a "description" we mean a phrase of the form "the 
so-and-so" or of some equivalent form. For the present, we confine our 
attention to the in the singular. We shall use this word strictly, so as to 
imply uniqueness ; e.g. we should not say "A is the son of B " if B had other 
sons besides A. Thus a description of the form "the so-and-so" will only 
have an application in the event of there being one so-and-so and no more. 
Hence a description requires some propositional function <f>£ which is satisfied 
by one value of x and by no other values ; then " the x which satisfies <f>£ " 
is a description which definitely describes a certain object, though we may 
not know what object it describes. For example, if y is a man, "x is the 
father of y " must be true for one, and only one, value of x. Hence " the 
father of y " is a description of a certain man, though we may not know what 
man it describes. A phrase containing "the" always presupposes some initial 
propositional function not containing "the"; thus instead of "x is the father 
of y " we ought to take as our initial function " x begot y " ; then " the father 
of y " means the one value of x which satisfies this propositional function. 

If <f>& is a propositional function, the symbol "(ix)(<j>x)" is used in our 
symbolism in such a way that it can always be read as " the x which satisfies 
<f>$." But we do not define " (ix) (<f>x) " as standing for " the x which satisfies 
<£&," thus treating this last phrase as embodying a primitive idea. Every use 
of " (ix) (<f)x)," where it apparently occurs as a constituent of a proposition 
in the place of an object, is defined in terms of the primitive ideas already 
on hand. An example of this definition in use is given by the proposition 
" E ! (ix)(<px) " which is considered immediately. The whole subject is treated 
more fully in Chapter III. 

The symbol should be compared and contrasted with " & ((f>x) " which in 
use can always be read as "the x's which satisfy <f>&." Both symbols are in- 
complete symbols defined only in use, and as such are discussed in Chapter III. 
The symbol " cb (<f>x) " always has an application, namely to the class determined 
by <f>x ; but " (ix) (<f>x) " only has an application when </>£ is only satisfied by 
one value of x, neither more nor less. It should also be observed that the 
meaning given to the symbol by the definition, given immediately below, of 
E ! (ix) (<f>x) does not presuppose that we know the meaning of " one." This is 
also characteristic of the definition of any other use of (ix) (<f>x). 

We now proceed to define " E ! (ix) (<px) " so that it can be read " the x 
satisfying <f>x exists." (It will be observed that this is a different meaning of 
existence from that which we express by " g.") Its definition is 
E ! (ix) (<f>x) . = : (gc) :<f>x.= x .x = c Df, 



I] DESCRIPTIONS 31 

i.e. " the x satisfying <f>$; exists " is to mean " there is an object c such that cf>x 
is true when x is c but not otherwise." 

The following are equivalent forms : 

H :. E! (ix) (<f)x) . = : (gc) : <f>c : <f>x . D x . x = c, 

H :. E! (ix) (<f>x) . = : (gc) . <f>c : <f>x . <f>y . D XjV .x = y, 

\- :. E ! (7a?) ($#) . = : (gc) : <f>c : a? 4= c . D x ■ ~ <£#• 

The last of these states that " the x satisfying <f>£ exists " is equivalent to 
"there is an object c satisfying <f>$!, and every object other than c does not 
satisfy </>&." 

The kind of existence just defined covers a great many cases. Thus for 
example " the most perfect Being exists " will mean : 

(gc) : x is most perfect . = x . x = c, 
which, taking the last of the above equivalences, is equivalent to 
(gc) : c is most perfect : x =J= c . D^ . x is not most perfect. 

A proposition such as "Apollo exists " is really of the same logical form, 
although it does not explicitly contain the word the. For "Apollo" means 
really " the object having such-and-such properties," say " the object having 
the properties enumerated in the Classical Dictionary*." If these properties 
make up the propositional function <f>x, then "Apollo" means " (ix) (<f>x)," 
and "Apollo exists" means "E! (ix)(<f>x)." To take another illustration, 
" the author of Waverley" means " the man who (or rather, the object which) 
wrote Waverley." Thus " Scott is the author of Waverley " is 

Scott = (ix) (x wrote Waverley). 
Here (as we observed before) the importance of identity in connection with 
descriptions plainly appears. 

The notation " (ix) (<f>x)" which is long and inconvenient, is seldom used, 
being chiefly required to lead up to another notation, namely "R'y" meaning 
" the object having the relation R to y." That is, we put 

R'y = (ix) (xRy) Df. 
The inverted comma may be read "of." Thus "R'y" is read "the R of y." 
Thus if R is the relation of father to son, "R'y" means "the father of y"; 
if R is the relation of son to father, "R'y " means " the son of y," which will 
only " exist " if y has one son and no more. R'y is a function of y, but not 
a propositional function; we shall call it a descriptive function. All the 
ordinary functions of mathematics are of this kind, as will appear more fully 
in the sequel. Thus in our notation, " sin y " would be written " sin 'y" and 
" sin " would stand for the relation which sin 'y has to y. Instead of a variable 
descriptive function fy, we put R'y, where the variable relation R takes the 

* The same principle applies to many uses of the proper names of existent objects, e.g. to all 
nses of proper names for objects known to the speaker only by report, and not by personal 
acquaintance. 



32 INTRODUCTION [CHAP. 

place of the variable function /. A descriptive function will in general exist 
while y belongs to a certain domain, but not outside that domain ; thus if we 
are dealing with positive rationals, sjy will be significant if y is a perfect 
square, but not otherwise; if we are dealing with real numbers, and agree 
that " \]y " is to mean the positive square root (or, is to mean the negative 
square root), \'y will be significant provided y is positive, but not otherwise ; 
and so on. Thus every descriptive function has what we may call a " domain 
of definition " or a " domain of existence," which may be thus defined : If the 
function in question is R'y, its domain of definition or of existence will be 
the class of those arguments y for which we have E! R'y, i.e. for which 
E ! (ix) (xRy), i.e. for which there is one x, and no more, having the relation 
R to y. 

If R is any relation, we will speak of R'y as the " associated descriptive 
function." A great many of the constant relations which we shall have occasion 
to introduce are only or chiefly important on account of their associated descrip- 
tive functions. / In such cases, it is easier (though less correct) to begin by 
assigning the meaning of the descriptive function, and to deduce the meaning 
of the relation from that of the descriptive function. This will be done in the 
following explanations of notation. 

Various descriptive functions of relations. If R is any relation, the converse 
of R is the relation which holds between y and x whenever R holds between 
x and y. Thus greater is the converse of less, before of after, cause of effect 

husband of wife, etc. The converse of R is written* Cnv'R or R. The defi- 
nition is 

R = $$(yRx) Df, 

Cnv'R = R Df. 

The second of these is not a formally correct definition, since we ought to 
define " Cnv " and deduce the meaning of Cnv'R. But it is not worth while 
to adopt this plan in our present introductory account, which aims at simplicity 
rather than formal correctness. 

A relation is called symmetrical if R = R, i.e. if it holds between y and x 
whenever it holds between x and y (and therefore vice versa). Identity, 
diversity, agreement or disagreement in any respect, are symmetrical relations. 
A relation is called asymmetrical when it is incompatible with its converse, 

i.e. when R r\ R = A, or, what is equivalent, 

xRy . D x ,y ■ ~ (yRx). 

Before and after, greater and less, ancestor and descendant, are asym- 
metrical, as are all other relations of the sort that lead to series. But there are 
many asymmetrical relations which do not lead to series, for instance, that of 

* The second of these notations is taken from Schroder's Algebra und Logik der Relative. 



I] DESCRIPTIVE FUNCTIONS 33 

wife's brother*. A relation may be neither symmetrical nor asymmetrical ; 
for example, this holds of the relation of inclusion between classes : a C /3 and 
j3 C a will both/be true if a = /8, but otherwise only one of them, at most, will 
be true. The relation brother is neither symmetrical nor asymmetrical, for if 
x is the brother of y, y may be either the brother or the sister of x. 

In the prepositional function xRy, we call x the referent and y the relatum. 
The class x (xRy), consisting of all the xs which have the relation R to y, is 
called the class of referents of y with respect to R; the class $ (xRy), consisting 
of all the y'a to which x has the relation R x is called the class of relata of x 

with respect to R. These two classes are denoted respectively by R'y and R l x. 
Thus 

R'y = $(xRy) Df, 

R'x = §{xRy) Df. 
The arrow runs towards y in the first case, to show that we are concerned 
with things having the relation R to y; it runs away from x in the second 

case, to show that the relation R goes from x to the members of R l x. It runs 
in fact from a referent and towards a relatum. 

-* *- 

The notations R'y, R'x are very important, and are used constantly. If 

— * «— 

R is the relation of parent to child, R'y = the parents of y, R'x = the children 

of x. We have 

— ► 
h : x e R'y . = . xRy 

and I- : y e R'x . = . xRy. 

These equivalences are often embodied in common language. For example, 
we say indiscriminately u x is an inhabitant of London" or "x inhabits London." 
If we put "R"for "inhabits," "x inhabits London" is "xR London," while "x 

is an inhabitant of London " is " x e R' London." 
— > <— 
Instead of R and R we sometimes use sg'R, gs'R, where " sg " stands for 
" sagitta," and " gs " is " sg " backwards. Thus we put 

sg'R = R Df, 

gs'R = R Df. 
These notations are sometimes more convenient than an arrow when the 
relation concerned is represented by a combination of letters, instead of a 
single letter such as R. Thus e.g. we should write sg'(.R r\ S), rather than put 
an arrow over the whole length of (R r\ 8). 

The class of all terms that have the relation R to something or other is 
called the domain of R. Thus if R is the relation of parent and child, the 

* This relation is not strictly asymmetrical, but is so except when the wife's brother is also 
the sister's husband. In the Greek Church the relation is strictly asymmetrical. 

R&W I 3 



34 INTRODUCTION [CHAP. 

domain of R will be the class of parents. We represent the domain of R by 
" D'R." Thus we put 

D<E = £{( a y).*%} Df. 

Similarly the class of all terms to which something or other has the relation 
R is called the converse domain of R ; it is the same as the domain of the 
converse of R. The converse domain of R is represented by " CE'JR "; thus 

Q-'R = $ {(a«) ■ xR y) Df - 

The sum of the domain and the converse domain is called the field, and is 

represented by C'R: thus 

C'R = D'Ryj(l'R Df. 

The field is chiefly important in connection with series. If R is the ordering 
relation of a series, C'R will be the class of terms of the series, D' R will be all 
the terms except the last (if any), and (I'R will be all the terms except the 
first (if any). The first term, if it exists, is the only member of D'R n — Q.'R, 
since it is the only term which is a predecessor but not a follower. Similarly 
the last term (if any) is the only member of G.'R n - D'R. The condition 
that a series should have no end is Q'RCD'R, i.e. "every follower is a pre- 
decessor"; the condition for no beginning is D'R C (I'R. These conditions 
are equivalent respectively to D'R = C'R and d'R = C'R. 

The relative product of two relations R and S is the relation which holds 
between x and z when there is an intermediate term y such that x has the 
relation R to y and y has the relation S to z. The relative product of R and 
S is represented by R | S ; thus we put 

RlS^^K^.xRy.ySz} Df, 
whence h : x (R | S) z . = . (gy) . xRy . ySz. 

Thus "paternal aunt" is the relative product of sister and father; "paternal 
grandmother " is the relative product of mother and father ; " maternal grand- 
father " is the relative product of father and mother. The relative product is 
not commutative, but it obeys the associative law, i.e. 

\-.(P\Q)\R = P\(Q\R). 
It also obeys the distributive law with regard to the logical addition of 
relations, i.e. we have 

h.P\(QuR) = (P\Q)v(P\R), 

\-.(QvR)\P = (Q\P)v(R\P). 

But with regard to the logical product, we have only 
b.P\(QnR)G(P\Q)r.(P\R) > 
h.(QnR)\PG(Q\P)n(Q\R). 

The relative product does not obey the law of tautologjr, i, e . we do not 
have in general R R = R. We put 

R* = R\R Df. 



I] PLURAL DESCRIPTIVE FUNCTIONS 35 

Thus paternal grandfather = (father) 2 , 
maternal grandmother = (mother) 2 . 

A relation is called transitive when R 2 G R, i.e. when, if xRy and yRz, we 
always have xRz, i.e. when 

xRy . yRz . "^ x ,y,z • xRz. 
Relations which generate series are always transitive ; thus e.g. 

x>y .y> z t. D x , y , z ,x>z. 
If P is a relation which generates a series, P may conveniently be read 
"precedes"; thus "xPy .yPz .D Xt y Z .xPz" becomes "if a; precedes y and y 
precedes z, then x always precedes z." The class of relations which generate 
series are partially characterized by the fact that they are transitive and 
asymmetrical, and never relate a term to itself. 

If P is a relation which generates a series, and if we have not merely P 2 G P, 
but P 2 = P, then P generates a series which is compact (uberall dicht), i.e. such 
that there are terms between any two. For in this case we have 

xPz . D . (ay) . xPy . yPz, 
i.e. if x precedes z, there is a term y such that x precedes y and y precedes z, 
i.e. there is a term between x and z. Thus among relations which generate 
series, those which generate compact series are those for which P* — P. 

Many relations which do not generate series are transitive, for example, 
identity, or the relation of inclusion between classes. Such cases arise when 
the relations are not asymmetrical. Relations which are transitive and sym- 
metrical are an important class : they may be regarded as consisting in the 
possession of some common property. 

Plural descriptive functions. The class of terms x which have the relation 
R to some member of a class a is denoted by R"a or R/a. The definition is 

R"a = x-{(Ry).yea.xRy} Df. 
Thus for example let R be the relation of inhabiting, and a the class of towns; 
then R"a = inhabitants of towns. Let R be the relation " less than " among 
rationals, and a the class of those rationals which are of the form 1 — 2~ n , for 
integral values of n ; then R"a will be all rationals less than some member 
of a, i.e. all rationals less than 1. If P is the generating relation of a series, 
and a is any class of members of the series, P"a will be predecessors of as, i.e. the 
segment defined by a. If P is a relation such that P l y always exists when 
yea, P"a will be the class of all terms of the form P'y for values of y which 
are members of a ; i.e. 

P"a = ${(>&y).yea.x = P<y}. 

Thus a member of the class " fathers of great men " will be the father of y, 
where y is some great man. In other cases, this will not hold ; for instance, 
let P be the relation of a number to any number of which it is a factor ; then 

3—2 - 



36 INTRODUCTION [CHAP. I 

P" (even numbers) = factors of even numbers, but this class is not composed 
of terms of the form " the factor of x" where x is an even number, because 
numbers do not have only one factor apiece. 

Unit classes. The class whose only member is x might be thought to be 
identical with x, but Peano and Frege have shown that this is not the case. 
(The reasons why this is not the case will be explained in a preliminary way 
in Chapter II of the Introduction.) We denote by " i l x " the class whose only 
member is x : thus 

i'x = ${y = x) Df, 

i.e. "i'x" means "the class of objects which are identical with x." 

The class consisting of x and y will be i'x \j i l y ; the class got by adding 
a; to a class a will be aw l'x\ the class got by taking away x from a class a 
will be a — i'x. (We write a — ft as an abbreviation for a r\ — ft.) 

It will be observed that unit classes have been defined without reference 
to the number 1 ; in fact, we use unit classes to define the number 1. This 
number is defined as the class of unit classes, i.e. 

l = a{{<&x).a = l'x} Df. 
This leads to 

I- :. « e 1 . = : (&x) : y e a . = y . y =x: 

From this it appears further that 

r- : a e 1 . = . E ! (ix) {x e a), 
whence V : % (<t>z) e 1 . = . E ! (ix) (<f>x), 

i.e. " % (<f>z) is a unit class " is equivalent to " the x satisfying <f>fc exists." 

If a e 1, t'a is the only member of a, for the only member of a is the only 

term to which a has the relation i. Thus "iV takes the place of "(ix)(<f>x)" 

if a stands for z(<f>z). In practice, "t'a" is a more convenient notation than 
" (ix) (<f)x)," and is generally used instead of " (ix) (<f>x)." 

The above account has explained most of the logical notation employed 
in the present work. In the applications to various parts of mathematics, 
other definitions are introduced; but the objects defined by these later defi- 
nitions belong, for the most part, rather to mathematics than to logic. The 
reader who has mastered the symbols explained above will find that any 
later formulae can be deciphered by the help of comparatively few additional 
definitions. 



CHAPTER II 

THE THEOKY OF LOGICAL TYPES 

The theory of logical types, to be explained in the present Chapter, re- 
commended itself to us in the first instance by its ability to solve certain 
contradictions, of which the one best known to mathematicians is Burali-Forti's 
concerning the greatest ordinal. But the theory in question is not wholly 
dependent upon this indirect recommendation: it has also a certain consonance 
with common sense which makes it inherently credible. In what follows, we 
shall therefore first set forth the theory on its own account, and then apply it 
to the solution of the contradictions. 

I. The Vicious-Circle Principle. 

An analysis of the paradoxes to be avoided shows that they all result from 
a certain kind of vicious circle*. The vicious circles in question arise from 
supposing that a collection of objects may contain members which can only be 
defined by means of the collection as a whole. Thus, for example, the collection 
of propositions will be supposed to contain a proposition stating that " all 
propositions are either true or false." It would seem, however, that such a 
statement could not be legitimate unless "all propositions" referred to some 
already definite collection, which it cannot do if new propositions are created 
by statements about " all propositions." We shall, therefore, have to say that 
statements about "all propositions" are meaningless. More generally, given 
any set of objects such that, if we suppose the set to have a total, it will con- 
tain members which presuppose this total, then such a set cannot have a total. 
By saying, that a set has "no total," we mean, primarily, that no significant 
statement can be made about "all its members." Propositions, as the above 
illustration shows, must be a set having no total. The same is true, as we shall 
shortly see, of propositional functions, even when these are restricted to such 
as can significantly have as argument a given object a. In such cases, it is 
necessary to break up our set into smaller sets, each of which is capable of a 
total. This is what the theory of types aims at effecting. 

The principle which enables us to avoid illegitimate totalities may be 
stated as follows: "Whatever involves all of a collection must not be one of 
the collection"; or, conversely: "If, provided a certain collection had a total, 
it would have members only definable in terms of that total, then the said 
collection has no total." We shall call this the " vicious-circle principle," be- 
cause it enables us to avoid the vicious circles involved in the assumption of 
illegitimate totalities. Arguments which are condemned by the vicious-circle 

* See the last section of the present Chapter. Cf. also H. Poincar£, " Les math^matiques et 
la logique," Revue de Metaphysique et de Morale, Mai 1906, p. 307. 



38 INTRODUCTION [CHAP. 

principle will be called "vicious-circle fallacies." Such arguments, in certain 
circumstances, may lead to contradictions, but it often happens that the con- 
clusions to which they lead are in fact true, though the arguments are 
fallacious. Take, for example, the law of excluded middle, in the form " all 
propositions are true or false." If from this law we argue that, because the 
law of excluded middle is a proposition, therefore the law of excluded middle 
is true or false, we incur a vicious-circle fallacy. "All propositions" must be 
in some way limited before it becomes a legitimate totality, and any limita- 
tion which makes it legitimate must make any statement about the totality 
fall outside the totality. Similarly, the imaginary sceptic, who asserts that 
he knows nothing, and is refuted by being asked if he knows that he knows 
nothing, has asserted nonsense, and has been fallaciously refuted by an 
argument which involves a vicious-circle fallacy. In order that the sceptic's 
assertion may become significant, it is necessary to place some limitation 
upon the things of which he is asserting his ignorance, because the things 
of which it is possible to be ignorant form an illegitimate totality. But as 
soon as a suitable limitation has been placed by him upon the collection of 
propositions of which he is asserting his ignorance, the proposition that he is 
ignorant of every member of this collection must not itself be one of the 
collection. Hence any significant scepticism is not open to the above form of 
refutation. 

The paradoxes of symbolic logic concern various sorts of objects: propo- 
sitions., classes, cardinal and ordinal numbers, etc. All these sorts of objects, 
as we shall show, represent illegitimate totalities, and are therefore capable of 
giving rise to vicious-circle fallacies. But by means of the theory (to be 
explained in Chapter III) which reduces statements that are verbally con- 
cerned with classes and relations to statements that are concerned with 
propositional functions, the paradoxes are reduced to such as are concerned 
with propositions and propositional functions. The paradoxes that concern 
propositions are only indirectly relevant to mathematics, .while those that 
more nearly concern the mathematician are all concerned with propositional 
functions. We shall therefore proceed at once to the consideration of propo- 
sitional functions. 

II. The Nature of Propositional Functions. 
By a "propositional function" we mean something which contains a 
variable x, and expresses a proposition as soon as a value is assigned to x. 
That is to say, it differs from a proposition solely by the fact that it is 
ambiguous : it contains a variable of which the value is unassigned. It agrees 
with the ordinary functions of mathematics in the fact of containing an 
unassigned variable; where it differs is in the fact that the values of the 
function are propositions. Thus e.g. "x is a man" or "sin# = 1 " is a propo- 
sitional function. We shall find that it is possible to incur a vicious-circle 



II] PROPbSITIONAL FUNCTIONS 39 

fallacy at the very outset, by admitting as possible arguments to a propositional 
function terms which presuppose the function. This form of the fallacy is very 
instructive, and its avoidance leads, as we shall see, to the hierarchy of types. 

The question as to the nature of a function* is by no means an easy one. 
It would seem, however, that the essential characteristic of a function is 
ambiguity. Take, for example, the law of identity in the form "A is A" which 
is the form in which it is usually enunciated. It is plain that, regarded 
psychologically, we have here a single judgment. But what are we to say of 
the object of the judgment ? We are not judging that Socrates is Socrates, 
nor that Plato is Plato, nor any other of the definite judgments that are 
instances of the law of identity. Yet each of these judgments is, in a sense, 
within the scope of our judgment. We are in fact judging an ambiguous 
instance of the propositional function " A is A." We appear to have a single 
thought which does not have a definite object, but has as its object an 
undetermined one of the values of the function "A is A." It is this kind of 
ambiguity that constitutes the essence of a function. When we speak of "</>#," 
where x is not specified, we mean one value of the function, but not a definite 
one. We may express this by saying that "<f>x" ambiguously denotes <f>a, <f>b, <£c, 
etc., where <f>a, <f>b, <f>c, etc., are the various values of "<£#." 

When we say that "<f>x" ambiguously denotes <f>a, <$>b, </>c, etc., Ave mean 
that "<$>x" means one of the objects <f>a, <f)b, <f>c, etc., though not a definite 
one, but an undetermined one. It follows that "<f>x" only has a well-defined 
meaning (well-defined, that is to say, except in so far as it is of its essence to 
be ambiguous) if the objects $>a, <f>b, <£c, etc., are well-defined. That is to say, 
a function is not a well-defined function unless all its values are already well- 
defined. It follows from this that no function can have among its values 
anything which presupposes the function, for if it had, we could not regard 
the objects ambiguously denoted by the function as definite until the function 
was definite, while conversely, as we have just seen, the function cannot be 
definite until its values are definite. This is a particular case, but perhaps the 
most fundamental case, of the vicious-circle principle. A function is what 
ambiguously denotes some one of a certain totality, namely the values of the 
function ; hence this totality cannot contain any members which involve the 
function, since, if it did, it would contain members involving the totality, 
which, by the vicious-circle principle, no totality can do. 

It will be seen that, according to the above account, the values of a 

function are presupposed by the function, not vice versa. It is sufficiently 

obvious, in any particular case, that a value of a function does not presuppose 

the function. Thus for example the proposition " Socrates is human " can be 

perfectly apprehended without regarding it as a value of the function "x is 

human." It is true that, conversely, a function can be apprehended without 

* "When the word " function " is used in the sequel, "propositional function " is always meant. 
Other functions will not be in question in the present Chapter. 



40 INTRODUCTION [CHAP. 

its being necessary to apprehend its values severally and individually. If this 
were not the case, no function could be apprehended at all, since the number 
of values (true and false) of a function is necessarily infinite and there are 
necessarily possible arguments with which we are unacquainted. What is 
necessary is not that the values should be given individually and extensionally, 
but that the totality of the values should be given intensionally, so that, con- 
cerning any assigned object, it is at least theoretically determinate whether or 
not the said object is a value of the function. 

It is necessary practically to distinguish the function itself from an 
undetermined value of the function. We may regard the function itself as 
that which ambiguously denotes, while an undetermined value of the function 
is that which is ambiguously denoted. If the undetermined value is written 
"<f>x," we will write the function itself "</>&." (Any other letter may be used 
in place of x.) Thus we should say "<j)% is a proposition," but "<]>& is a prepo- 
sitional function." When we say "<f>x is a proposition," we mean to state 
something which is true for every possible value of x, though we do not decide 
what value x is to have. We are making an ambiguous statement about any 
value of the function. But when we say " <f)x is a function," we are not making 
an ambiguous statement. It would be more correct to say that we are making 
a statement about an ambiguity, taking the view that a function is an am- 
biguity. The function itself, <f>w, is the single thing which ambiguously denotes 
its many values ; while <f>x, where x is not specified, is one of the denoted 
objects, with the ambiguity belonging to the manner of denoting. 

We have seen that, in accordance with the vicious-circle principle, the 
values of a function cannot contain terms only definable in terms of the 
function. Now given a function <f>£, the values for the function* are all pro- 
positions of the form <px. It follows that there must be no propositions, of 
the form <px, in which x has a value which involves <j><jc. (If this were the case, 
the values of the function would not all be determinate until the function 
was determinate, whereas we found that the function is not determinate unless 
its values are previously determinate.) Hence there must be no such thing as 
the value for <f>Zc with the argument <£&, or with any argument which involves 
cf>x. That is to say, the symbol "<f> (<f>£)" must not express a proposition, as 
"<f>a" does if <f>a is a value for </>£. In fact "<f> (<f>£)" must be a symbol which 
does not express anything: we may therefore say that it is not significant. Thus 
given any function <j>x, there are arguments with which the function has no 
value, as well as arguments with which it has a value. We will call the 
arguments with which <px has a value "possible values of x." We will say 
that cf)x is "significant with the argument x" when <f>x has a value with the 
argument x. 

* We shall speak in this Chapter of "values for (f>x" and of "values of 4>x" meaning in eacli 
case the same thing, namely (pa, <f>b, <pc, etc. The distinction of phraseology serves to avoid 
ambiguity where several variables are concerned, especially when one of them is a function. 



II] POSSIBLE ARGUMENTS FOR FUNCTIONS 41 

When it is said that e.g. " <f> (<£2)" is meaningless, and therefore neither 
true nor false, it is necessary to avoid a misunderstanding. If "(f> (<f>2)" were 
interpreted as meaning "the value for $z with the argument <f>% is true," 
that would be not meaningless, but false. It is false for the same reason for 
which "the King of France is bald" is false, namely because there is no such 
thing as "the value for $z with the argument <f>1z." But when, with some 
argument a, we assert <pa, we are not meaning to assert "the value for <j>^c with 
the argument a is true"; we are meaning to assert the actual proposition 
which is the value for <f>x with the argument a. Thus for example if $» is 
"x is a man," <f> (Socrates) will be "Socrates is a man," not "the value for 
the function l x is a man/ with the argument Socrates, is true." Thus 
in accordance with our principle that "</> (#2)" is meaningless, we cannot 
legitimately deny "the function 'x is a man' is a man," because this is 
nonsense, but we can legitimately deny "the value for the function 'x is a 
man' with the argument 'x is a man' is true," not on the ground that the 
value in question is false, but on the ground that there is no such value for 
the function. 

We will denote by the symbol "(x) . <f>x" the proposition "<f>x always*," 
i.e. the proposition which asserts all the values for <f>&. This proposition 
involves the function <f>£, not merely an ambiguous value of the function. The 
assertion of <f>x, where x is unspecified, is a different assertion from the one 
which asserts all values for <f>x~, for the former is an ambiguous assertion, 
whereas the latter is in no sense ambiguous. It will be observed that "(x).<f)x" 
does not assert "<f>x with all values of x" because, as we have seen, there must 
be values of x with which "<f>x" is meaningless. What is asserted by "(x).<f)x" 
is all propositions which are values for <f>£ ; hence it is only with such values 
of x as make "<f>x" significant, i.e. with all possible arguments, that <f>x is asserted 
when we assert "(x) . <f>x." Thus a convenient way to read "(x) . <f>x" is "<f>x is 
true with all possible values of x." This is, however, a less accurate reading 
than "<j>x always," because the notion of truth is not part of the content of 
what is judged. When we judge "all men are mortal," we judge truly, but 
the notion of truth is not necessarily in our minds, any more than it need be- 
when we judge "Socrates is mortal." 

III. Definition and Systematic Ambiguity of Truth and Falsehood. 

Since "(x) . <j>x" involves the function </>&, it must, according to our 
principle, be impossible as an argument to <f>. That is to say, the symbol 
"<f> [(x) . <j>x}" must be meaningless. This principle would seem, at first sight, 
to have certain exceptions. Take, for example, the function "p is false," and 
consider the proposition "(p) . p is false." This should be a proposition 
asserting all propositions of the form " p is false." Such a proposition, we 

* We use "always" as meaning "in all cases," not "at all times." Similarly "sometimes" 
will mean ' * in some cases." 



42 INTRODUCTION [CHAP. 

should be inclined to say, must be false, because "p is false" is not always 
true. Hence we should be led to the proposition 

" {(p) ■ P i s ^ se ] is false," 
i.e. we should be led to a proposition in which "(p) .p is false" is the argu- 
ment to the function "p is false," which we had declared to be impossible. 
Now it will be seen that "(p).p is false," in the above, purports to be a 
proposition about all propositions, and that, by the general form of the vicious- 
circle principle, there must be no propositions about all propositions. Never- 
theless, it seems plain that, given any function, there is a proposition (true or 
false) asserting all its values. Hence we are led to the conclusion that "p is 
false" and "q is false" must not always be the values, with the arguments p 
and q, for a single function " p is false." This, however, is only possible if the 
word "false" really has many different meanings, appropriate to propositions 
of different kinds. 

That the words "true" and "false" have many different meanings, accord- 
ing to the kind of proposition to which they are applied, is not difficult to 
see. Let us take any function <f>x, and let <f>a be one of its values. Let us call 
the sort of truth which is applicable to </>a "first truth." (This is not to assume 
that this would be first truth in another context: it is merely to indicate that 
it is the first sort of truth in our context.) Consider now the proposition 
(x) . <f>x. If this has truth of the sort appropriate to it, that will mean that 
every value <f>x has "first truth." Thus if we call the sort of truth that is 
appropriate to (x) . <f>x " second truth," we may define "{{x).<j>x} has second 
truth" as meaning "every value for <f>x has first truth," i.e. "(x) . {<f>% has first 
truth)." Similarly, if we denote by "(ga:) . <f>x" the proposition "<j>x sometimes," 
i.e. as we may less accurately express it, "(/># with some value of x," we find 
that (gav) . (f)x has second truth if there is an x with which <f>x has first truth ; 
thus we may define " {(gar) . <£a?} has second truth" as meaning "some value 
for <f>x has first truth," i.e. "(g#) • ($x has first truth)." Similar remarks apply 
to falsehood. Thus "{(x).<px) has second falsehood" will mean "some value 
for <j>$ has first falsehood," i.e. "(qx) . (</># has first falsehood)," while 
" {(a#) . 4>x) has second falsehood" will mean "all values for $$ have first 
falsehood," i.e. "(x) . (cf)X has first falsehood)." Thus the sort of falsehood that 
can belong to a general proposition is different from the sort that can belong 
to a particular proposition. 

Applying these considerations to the proposition "(p) . p is false," we see 
that the kind of falsehood in question must be specified. If, for example, 
first falsehood is meant, the function "p has first falsehood" is only signi- 
ficant when p is the sort of proposition which has first falsehood or first 
truth. Hence "{p).p is false" will be replaced by a statement which is 
equivalent to "all propositions having either first truth or first falsehood 
have first falsehood." This proposition has second falsehood, and is not 



II] TRUTH AND FALSEHOOD 43 

a possible argument to the function "p has first falsehood." Thus the 
apparent exception to the principle that "<f> {(#) . <j>x}" must be meaningless 
disappears. 

Similar considerations will enable us to deal with "not-p" and with "p or q." 
It might seem as if these were functions in which any proposition might 
appear as argument. But this is due to a systematic ambiguity in the mean- 
ings of "not" and "or," by which they adapt themselves to propositions of any 
order. To explain fully how this occurs, it will be well to begin with a 
definition of the simplest kind -of truth and falsehood. 

The universe consists of objects having various qualities and standing 
in various relations. Some of the objects which occur in the universe are 
complex. When an object is complex, it consists of interrelated parts. Let 
us consider a complex object composed of two parts a and b standing to each 
other in the relation R. The complex object "a-in-the-relation-jR-to-6" may 
be capable of being perceived ; when perceived, it is perceived as one object. 
Attention may show that it is complex ; we then judge that a and b stand in 
the relation R. Such a judgment, being derived from perception by mere 
attention, may be called a "judgment of perception." This judgment of 
perception, considered as an actual occurrence, is a relation of four terms, 
namely a and b and R and the percipient. The perception, on the contrary, is 
a relation of two terms, namely "a-in-the-relation-i2-to-6," and the percipient. 
Since an object of perception cannot be nothing, we cannot perceive "a-in-the- 
relation-i?-to-6 " unless a is in the relation jR to b. Hence a judgment of 
perception, according to the above definition, must be true. This does not 
mean that, in a judgment which appears to us to be one of perception, we 
are sure of not being in error, since we may err in thinking that our judgment 
has really been derived merely by analysis of what was perceived. But if our 
judgment has been so derived, it must be true. In fact, we may define truth, 
where such judgments are concerned, as consisting in the fact that there is a 
complex corresponding to the discursive thoughtwhich is the judgment. That is, 
when we judge "a has the relation R to b" our judgment is said to be true 
when there is a complex "<x-in-the-relation-JS-to-6," and is said to be false 
when this is not the case. This is a definition of truth and falsehood in rela- 
tion to judgments of this kind. 

It will be seen that, according to the above account, a judgment does not 
have a single object, namely the proposition, but has several interrelated 
objects. That is to say, the relation which constitutes judgment is not a 
relation of two terms, namely the judging mind and the proposition, but is a 
relation of several terms, namely the mind and what are called the constituents 
of the proposition. That is, when we judge (say) "this is red," what occurs 
is a relation of three terms, the mind, and "this," and red. On the other hand, 
when we perceive "the redness of this," there is a relation of two terms, namely 



44 INTRODUCTION [CHAP. 

the mind and the complex object "the redness of this." When a judgment 
occurs, there is a certain complex entity, composed of the mind and the 
various objects of the judgment. When the judgment is true, in the case of 
the kind of judgments we have been considering, there is a corresponding 
complex of the objects of the judgment alone. Falsehood, in regard to our 
present class of judgments, consists in the absence of a corresponding complex 
composed of the objects alone. It follows from the above theory that a 
"proposition," in the sense in which a proposition is supposed to be the object 
of a judgment, is a false abstraction, because a judgment has several objects, 
not one. It is the severalness of the objects in judgment (as opposed to 
perception) which has led people to speak of thought as "discursive," though 
they do not appear to have realized clearly what was meant by this epithet. 

Owing to the plurality of the objects of a single judgment, it follows that 
what we call a "proposition" (in the sense in which this is distinguished from 
the phrase expressing it) is not a single entity at all. That is to say, the phrase 
which expresses a proposition is what we call an "incomplete" symbol*; it 
does not have meaning in itself, but requires some supplementation in order 
to acquire a complete meaning. This fact is somewhat concealed by the 
circumstance that judgment in itself supplies a sufficient supplement, and that 
judgment in itself makes no verbal addition to the proposition. Thus "the 
proposition 'Socrates is human"' uses "Socrates is human" in a way which 
requires a supplement of some kind before it acquires a complete meaning; 
but when I judge "Socrates is human," the meaning is completed by the act of 
judging, and we no longer have an incomplete symbol. The fact that propositions 
are "incomplete symbols" is important philosophically, and is relevant at certain 
points in symbolic logic. 

The judgments we have been dealing with hitherto are such as are of the 
same form as judgments of perception, i.e. their subjects are always particular 
and definite. But there are many judgments which are not of this form. Such 
are "all men are mortal," "I met a man," "some men are Greeks." Before 
dealing with such judgments, we will introduce some technical terms. 

We will give the name of "a complex" to any such object as "a in the re- 
lation R to b" or "a having the quality q" or "a and b and c standing in the 
relation S." Broadly speaking, a complex is anything which occurs in the 
universe and is not simple. We will call a judgment elementary when it 
merely asserts such things as " a has the relation R to b," " a has the quality q " 
or "a and b and c stand in the relation S." Then an elementary judgment is 
true when there is a corresponding complex, and false when there is no corre- 
sponding complex. 

But take now such a proposition as "all men are mortal." Here the 
judgment does not correspond to one complex, but to many, namely "Socrates 

* See Chapter III. 



II] GENEBAL JUDGMENTS 45 

is mortal," "Plato is mortal," "Aristotle is mortal," etc. (For the moment, it 
is unnecessary to inquire whether each of these does not require further 
treatment before we reach the ultimate complexes involved. For purposes of 
illustration, "Socrates is mortal" is here treated as an elementary judgment, 
though it is in fact not one, as will be explained later. Truly elementary 
judgments are not very easily found.) We do not mean to deny that there 
may be some relation of the concept man to the concept mortal which may be 
equivalent to "all men are mortal," but in any case this relation is not the 
same thing as what we affirm when we say that all men are mortal. Our 
judgment that all men are mortal collects together a number of elementary 
judgments. It is not, however, composed of these, since {e.g.) the fact that 
Socrates is mortal is no part of what we assert, as may be seen by considering 
the fact that our assertion can be understood by a person who has never heard 
of Socrates. In prder to understand the judgment "all men are mortal," it is 
not necessary to know what men there are. We must admit, therefore, as a 
radically new kind of judgment, such general assertions as "all men are mortal." 
We assert that, given that x is human, x is always mortal. That is, we assert 
"x is mortal" of every x which is human. Thus we are able to judge (whether 
truly or falsely) that all the. objects which have some assigned property also 
have some other assigned property. That is, given any propositional functions 
<f>^e and yjr^c, there is a judgment asserting yfrx with every x for which we have 
<f>x. Such judgments we will call general judgments. 

It is evident (as explained above) that the definition of truth. is different 
in the case of general judgments from what it was in the case of elementary 
judgments. Let us call the meaning of truth which we gave for elementary 
judgments "elementary truth." Then when we assert that it is true that all 
men are mortal, we shall mean that all judgments of the form "x is mortal," 
where x is a man, have elementary truth. We may define this as "truth of 
the second order" or "second-order truth." Then if we express the proposition 
"air men are mortal" in the form 

"(x) . x is mortal, where a? is a man," 
and call this judgment p, then "p is true" must be taken to mean "p has 
second-order truth," which in turn means 

"(x) . 'x is mortal' has elementary truth, where a; is a man." 

In order to avoid the necessity for stating explicitly the limitation to 
which our variable is subject, it is convenient to replace the above interpre- 
tation of "all men are mortal" by a slightly different interpretation. The 
proposition "all men are mortal" is equivalent to "'x is a man' implies 'x is 
mortal,' with all possible values of #." Here x is not restricted to such values 
as are men, but may have any value with which "'x is a man' implies 'a; is 
mortal' " is significant, i.e. either true or false. Such a proposition is called a 
" formal implication." The advantage of this form is that the values which the 
variable may take are given by the function to which it is the argument: the 



46 INTRODUCTION [CHAP. 

values which the variable may take are all those with which the function is 
significant. 

We use the symbol "(x).<f>x" to express the general judgment which 
asserts all judgments of the form "<f>x." Then the judgment "all men are 
mortal" is equivalent to 

"(x) . \x is a man' implies 'x is a mortal,'" 
i.e. (in virtue of the definition of implication) to 

"(x) . x is not a man or x is mortal." 
As we have just seen, the meaning of truth which is applicable to this pro- 
position is not the same as the meaning of truth which is applicable to "# is a 
man" or to "x is mortal." And generally, in any judgment (x) . <f>x, the sense 
in which this judgment is or may be true is not the same as that in which <f>x 
is or may be true. If <f>x is an elementary judgment, it is true when it points 
to a corresponding complex. But (x) . <f>x does not point to a single corre- 
sponding complex : the corresponding complexes are as numerous as the possible 
values of x. 

It follows from the above that such a proposition as "all the judgments 
made by Epimenides are true" will only be prima facie capable of truth if all 
his judgments are of the same order. If they are of varying orders, of which 
the nth is the highest, we may make n assertions of the form "all the judg- 
ments of order m made by Epimenides are true," where m has all values up 
to n. But no such judgment can include itself in its own scope, since such a 
judgment is always of higher order than the judgments to which it refers. 

Let us consider next what is meant by the negation of a proposition of 
the form "(a?) . $x." We observe, to begin with, that "<f>x in some cases," or 
"<f>x sometimes," is a judgment which is on a par with "<f>x in all cases," or 
"<f>x always." The judgment "<f>x sometimes" is true if one or more values of 
x exist for which <f>x is true. We will express the proposition "<f>x sometimes" 
by the notation "(qx) .<f>x," where "g" stands for "there exists," and the 
whole symbol may be read "there exists an x such that <f>x." We take the 
two kinds of judgment expressed by "(x) . <j>x" and "(g#) . <f>x" as primitive 
ideas. We also take as a primitive idea the negation of "an elementary pro- 
position. We can then define the negations of (x) . (f>x and (gp) . <f>x. The 
negation of any proposition p will be denoted by the symbol "~p." Then the 
negation of (x) . <f>x will be defined as meaning 

"( a #).~<K' 
and the negation of (gp?) . <f>x will be defined as meaning "(x) . ~ <f>x." Thus, 
in the traditional language of formal logic, the negation of a universal affir- 
mative is to be defined as the particular negative, and the negation of the 
particular affirmative is to be defined as the universal negative. Hence the 
meaning of negation for such propositions is different from the meaning of 
negation for elementary propositions. 



II] SYSTEMATIC AMBIGUITY 47 

An analogous explanation will apply to disjunction. Consider the state- 
ment "either p, or <f>x always." We will denote the disjunction of two 
propositions p, q by "p v q." Then our statement is "p . v . (x) . <j>x." We will 
suppose that p is an elementary proposition, and that $x is always an elemen- 
tary proposition. We take the disjunction of two elementary propositions as 
a primitive idea, and we wish to define the disjunction 

"p . v . (x) . <f)X." 
This may be defined as "(x) .pv <j>x" i.e. "either p is true, or <f>x is always true" 
is to mean " 'p or <f>x' is always true." Similarly we will define 

"P • v • (3^) ■ $ x " 
as meaning "(g#) .p v <f>x," i.e. we define "either p is true or there is an x 
for which <f>x is true" as meaning "there is an x for which either p or (f>x is 
true." Similarly we can define a disjunction of two universal propositions: 
"( x ) • $ x ■ v . (y) . yfry" will be defined as meaning "(x,y) . <f>x v-tyy," i.e. 
"either <f>x is always true or yfry is always true" is to mean '"<f)X or tyy' is 
always true." By this method we obtain definitions of disjunctions con- 
taining propositions of the form (x) . (f>x or (ftx) . <j>x in terms of disjunctions 
of elementary propositions; but the meaning of "disjunction" is not the same 
for propositions of the forms (x) . <f>x. (qx) . <f>x, as it was for elementary pro- 
positions. 

Similar explanations could be given for implication and conjunction, but 
this is unnecessary, since these can be defined in terms of negation and 
disjunction. 

IV. Why a Given Function requires Arguments of a Certain Type. 
The considerations so far adduced in favour of the view that a function 
cannot significantly have as argument anything defined in terms of the 
function itself have been more or less indirect. But a direct consideration 
of the kinds of functions which have functions as arguments and the kinds 
of functions which have arguments other than functions will show, if we are 
not mistaken, that not only is it impossible for a function <f>z to have itself 
or anything derived from it as argument; but that, if yjr^ is another function 
such that there are arguments a with which both "<f>a" and "yfra" are sig- 
nificant, then yfr$ and anything derived from it cannot significantly be 
argument to $z. This arises from the fact that a function is essentially 
an ambiguity, and that, if it is to occur in a definite proposition, it must 
occur in such a way that the ambiguity has disappeared, and a wholly 
unambiguous statement has resulted. A few illustrations will make this clear. 
Thus "(x) . <f)x," which we have already considered, is a function of <j>£; as soon 
as <£& is assigned, we have a definite proposition, wholly free from ambiguity. 
But it is obvious that we cannot substitute for the function something which 
is not a function: "(x).<f)x" means "<£# in all cases," and depends for its 
significance upon the fact that there are "cases" of <f>%, i.e. upon the 



48 INTRODUCTION [CHAP. 

ambiguity which is characteristic of a function. This instance illustrates 
the fact that, when a function can occur significantly as argument, something 
which is not a function cannot occur significantly as argument. But con- 
versely, when something which is not a function can occur significantly 
as argument, a function cannot occur significantly. Take, e.g. "x is a man," 
and consider " <f>$ is a man," Here there is nothing to eliminate the 
ambiguity which constitutes* <f>x; there is thus nothing definite which is 
said to be a man. A function, in fact, is not a definite object, which could 
be or not be a man; it is a mere ambiguity awaiting determination, and 
in order that it may occur significantly it must receive the necessary deter- 
mination, which it obviously does not receive if it is merely substituted 
for something determinate in a proposition*. This argument does not, how- 
ever, apply directly as against such a statement as "{(x). <f>x] is a man." 
Common sense would pronounce such a statement to be meaningless, but it 
cannot be condemned on the ground of ambiguity in its subject. We need 
here a new objection, namely the following: A proposition is not a single entity, 
but a relation of several; hence a statement in which a proposition appears 
as subject will only be significant if it can be reduced to a statement about 
the terms which appear in the proposition. A proposition, like such phrases 
as "the so-and-so," where grammatically it appears as subject, must be broken 
up into its constituents if we are to find the true subject or subjects f. But 
in such a statement as "p is a man," where p is a proposition, this is not 
possible. Hence "{(x) . <£#} is a man" is meaningless. 

V. The Hierarchy of Functions and Propositions. 
We are thus led to the conclusion, both from the vicious-circle principle 
and from direct inspection, that the functions to which a given object a can 
be an argument are incapable of being arguments to each other, and that they 
have no term in common with the functions to which they can be arguments. 
We are thus led to construct a hierarchy. Beginning with a and the other 
terms which can be arguments to the same functions to which a can be argu- 
ment, we come next to functions to which a is a possible argument, and then 
to functions to which such functions are possible arguments, and so on. But 
the hierarchy which has to be constructed is not so simple as might at first 
appear. The functions which can take a as argument form an illegitimate 
totality, and themselves require division into a hierarchy of functions. This 
is easily seen as follows. Let f (<j>z, x) be a function of the two variables $z 
and x. Theii if, keeping x fixed for the moment, we assert this with all possible 
values of <f>, we obtain a proposition : 

(<f>).f(^,x). 

* Note that statements concerning the significance of a phrase containing "02" concern the 
symbol "<p2," and therefore do not fall under the rule that the elimination of the functional 
ambiguity is necessary to significance. Significance is a property of signs. Cf. pp. 40, 41. 

t Cf . Chapter III. 



n ] THE HIERARCHY OF FUNCTIONS 49 

Here, if x is variable, we have a function of x; but as this function involves 
a totality of values of <pz*, it cannot itself be one of the values included in 
the totality, by the vicious-circle principle. It follows that the totality of values 
of 02 concerned in (<f>) . f (<f>2, x) is not the totality of all functions in which 
x can occur as argument, and that there is no such totality as that of all func- 
tions in which x can occur as argument. 

It follows from the above that a function in which <f>2 appears as argument 
requires that "$$" should not stand for any function which is capable of a 
given argument, but must be restricted in such a way that none of the 
functions which are possible values of "<f>2" should involve any reference to 
the totality of such functions. Let us take as an illustration the definition 
of identity. We might attempt to define "x is identical with y" as meaning 
"whatever is true of x is true of y," i.e. "<f>x always implies (f>y." But here, 
since we are concerned to assert all values of "<j>x implies <f>y " regarded as a 
function of <f>, we shall be compelled to impose upon <j> some limitation which 
will prevent us from including among values of <f> values in which "all possible 
values of cf>" are referred to. Thus for example "x is identical with a" is a 
function of x; hence, if it is a legitimate value of <f> in "<f>x always implies 
<f>y" we shall be able to infer, by means of the above definition, that if x is 
identical with a, and x is identical with y, then y is identical with a. 
Although the conclusion is sound, the reasoning embodies a vicious-circle 
fallacy, since we have taken "(<f>) . <f>x implies <f>a" as a possible value of <f>x, 
which it cannot be. If, however, we impose any limitation upon <f>, it may 
happen, so far as appears at present, that with other values of $ we might 
have <f>x true and <f>y false, so that our proposed definition of identity would 
plainly be wrong. This difficulty is avoided by the "axiom of reducibility," 
to be explained later. For the present, it is only mentioned in order to 
illustrate the necessity and the relevance of the hierarchy of functions of a 
given argument. 

Let us give the name "a-functions" to functions that are significant for a 
given argument a. Then suppose we take any selection of a-functions, and 
consider the proposition "a satisfies all the functions belonging to the selection 
in question." If we here replace a by a variable, we obtain an a-function; but 
by the vicious-circle principle this a-function cannot be a member of our 
selection, since it refers to the whole of the selection. Let the selection consist 
of all those functions which satisfy/ (<f>z). Then our new function is 

((f)) . {f(<fiz) implies <f>x}, 
where x is the argument. It thus appears that, whatever selection of 
a-functions we may make, there will be other a-functions that lie outside our 

* When we speak of "values of <p$" it is <p, not z, that is to be assigned. This follows from 
the explanation in the note on p. 40. When the function itself is the variable, it is possible and 
simpler to write <j> rather than <f>z, except in positions where it is necessary to emphasize that an 
argument must be supplied to secure significance. 

R&W I 4 



50 INTRODUCTION [CHAP. 

selection. Such <z-functions, as the above instance illustrates, will always 
arise through taking a function of two arguments, <j>% and x, and asserting all 
or some of the values resulting from varying <f>. What is necessary, therefore, 
in order to avoid vicious-circle fallacies, is to divide our a-functions into 
"types," each of which contains no functions which refer to the whole of that 
type. 

When something is asserted or denied about all possible values or about 
some (undetermined) possible, values of a variable, that variable is called 
apparent, after Peano. The presence of the words all or some in a proposition 
indicates the presence of an apparent variable ; but often an apparent variable 
is really present where language does not at once indicate its presence. Thus 
for example "A is mortal" means "there is a time at which A will die," Thus 
a variable time occurs as apparent variable. 

The clearest instances of propositions not containing apparent variables 
are such as express immediate judgments of perception, such as "this is red" 
or "this is painful," where "this" is something immediately given. In other 
judgments, even where at first sight no variable appears to be present, it 
often happens that there really is one. Take (say) "Socrates is human." To 
Socrates himself, the word "Socrates" no doubt stood for an object of which 
he was immediately aware, and the judgment "Socrates is human" contained 
no apparent variable. But to us, who only know Socrates by description, the 
word "Socrates" cannot mean what it meant to him; it means rather "the 
person having such-and-such properties," (say) " the Athenian philosopher who 
drank the hemlock." Now in all propositions about "the so-and-so" there is 
an apparent variable, as will be shown in Chapter III. Thus in what we have 
in mind when we say "Socrates is human" there is an apparent variable, 
though there was no apparent variable in the corresponding judgment as 
made by Socrates, provided we assume that there is such a thing as immediate 
awareness of oneself. 

Whatever may be the instances of propositions not containing apparent 
variables, it is obvious that propositional functions whose values do not contain 
apparent variables are the source of propositions containing apparent variables, 
in the sense in which the function <f>£ is the source of the proposition (x) . $oc. 
For the values for <f>^ do not contain the apparent variable x, which appears 
in (x).<j>x; if they contain an apparent variable y, this can be similarly 
eliminated, and so on. This process must come to an end, since no proposition 
which we can apprehend can contain more than a finite number of apparent 
variables, on the ground that whatever we can apprehend must be of finite 
complexity. Thus we must arrive at last at a function of as many variables 
as there have been stages in reaching it from our original proposition, and 
this function will be such that its values contain no apparent variables. We 
may call this function the matrix of our original proposition and of any other 



II] MATBICES 51 

propositions and functions to be obtained by turning some of the arguments 
to the function into apparent variables. Thus for example, if we have a matrix- 
function whose values are ^> (x, y), we shall derive from it 
(y) . <f> (x, y), which is a function of x, 
(x) . <]> (x,y), which is a function of y, 
(x, y) . <£ (x, y), meaning "<f> (x, y) is true with all possible values of x and y." 
This last is a proposition containing no real variable, i.e. no variable except 
apparent variables. 

It is thus plain that all possible propositions and functions are obtainable 
from matrices by the process of turning the arguments to the matrices into 
apparent variables. In order to divide our propositions and functions into types, 
we shall, therefore, start from matrices, and consider how they are to be divided 
with a view to the avoidance of vicious-circle fallacies in the definitions of the 
functions concerned. For this purpose, we will use such letters as a, b, c, x, y, z, w, 
to denote objects which are neither propositions nor functions. Such objects 
we shall call individuals. Such objects will be constituents of propositions or 
functions, and will be genuine constituents, in the sense that they do not 
disappear on analysis, as (for example) classes do, or phrases of the form "the 
so-and-so." 

The first matrices that occur are those whose values are of the forms 
4>Xy^{x,y\x{oD,y,z...), 
i.e. where the arguments, however many there may be, are all individuals. 
The functions <f), tjr, %..., since (by definition) they contain no apparent 
variables, and have no arguments except individuals, do not presuppose any 
totality of 'functions. From the functions ^r, ^ ... we may proceed to form 
other functions of x, such as (y) . yjr (x, y), (gy) . ty (x, y), (y, z) . % {«>, y, z), 
(y) '• (3^) ■ X ( x > V> ?)> an( ^ so on - -^^ these presuppose no totality except that 
of individuals. We thus arrive at a certain collection of functions of x, 
characterized by the fact that they involve no variables except individuals. 
Such functions we will call "first-order functions." 

We may now introduce a notation to express "any first-order function." 
We will denote any first-order function by "<f>\&" and any value for such a 
function by "<f> ! x." Thus "<f> I x" stands for any value for any function which 
involves no variables except individuals. It will be seen that "<f> I x" is itself 
a function of two variables, namely <£ Vz and x. Thus <f> I x involves a variable 
which is not an individual, namely <£ ! z. Similarly "(x) . <f> ! x" is a function 
of the variable <j> 1 1, and thus involves a variable other than an individual. 
Again, if a is a given individual, 

"<f>lx implies (f> ! a with all possible values of <f>" 
is a function of x, but it is not a function of the form ^> ! x, because it involves 
an (apparent) variable <f> which is not an individual. Let us give the name 
"predicate" to any first-order function <f> I tb. (This use of the word "predicate" 

4—2 



52 INTRODUCTION [CHAP, 

is only proposed for the purposes of the present discussion.) Then the state- 
ment "<f>lx implies <f) I a with all possible values of <f>" may be read "all the 
predicates of x are predicates of a." This makes a statement about #,but does 
not attribute to # a predicate in the special sense just defined. 

Owing to the introduction of the variable first-order function 0!.t, we 
now have a new set of matrices. Thus "^> ! x" is a function which contains no 
apparent variables, but contains the two real variables cf> ! z and x. (It should 
be observed that when <f> is assigned, we may obtain a function whose values do 
involve individuals as apparent variables, for example if <f> I x is (y) . yfr (x, y). 
But so long as <£ is variable, <f) ! x contains no apparent variables.) Again, 
if a is a definite individual, <j> ! a is a function of the one variable <f> ! 2. 
If a and b are definite individuals, "<]>la implies yfr ! b" is a function of the 
two variables <f> ! 2, yfr ! t, and so on. We are thus led to a whole set of new 
matrices, 

/ (<j> ! z), g (<t> I % "f ! 2), F{<f> I % x), and so on. 

These matrices contain individuals and first-order functions as arguments, but 
(like all matrices) they contain no apparent variables. Any such matrix, if it 
contains more than one variable, gives rise to new functions of one variable 
by turning all its arguments except one into apparent variables. Thus we 
obtain the functions 

(<f>) .g(<$>\z, ">jrlz), which is a function of yjr I z. 

(x) ,F(<j>lz, x\ which is a function of <£ ! z. 

(0) .F((f>lz, x), which is a function of x. 

We will give the name of second-order matrices to such matrices as have 
first-order functions among their arguments, and have no arguments except 
first-order functions and individuals. (It is not necessary that they should 
have individuals among their arguments.) We will give the name of second- 
order functions to such as either are second-order matrices or are derived from 
such matrices by turning some of the arguments into apparent variables. It 
will be seen that either an individual or a first-order function may appear as 
argument to a second-order function. Second-order functions are such as con- 
tain variables which are first-order functions, but contain no other variables 
except (possibly) individuals. 

We now have various new classes of functions at our command. In the first 
place, we have second-order functions which have one argument which is a 
first-order function. We will denote a variable function of this kind by the 
notation f\ (<£ ! z), and any value of such a function by f\ (<£ ! z). Like 
<j> ! x, fl(tf>l z) is a function of two variables, namely f\ (<£ ! z) and <j> I z. Among 
possible values of /!(</>!£) will be <f> ! a (where a is constant), (x).<j>lx, 
(3#) .<j>lx, and so on. (These result from assigning a value to /, leaving 
$ to be assigned.) We will call such functions "predicative functions of 
first-order functions." 



II] SECOND-ORDER FUNCTIONS 53 

In the second place, we have second-order functions of two arguments, one 
of which is a first-order function while the other is an individual. Let us denote 
undetermined values of such functions by the notation 

As soon as x is assigned, we shall have a predicative function of <j> ! z. If our 
function contains no first-order function as apparent variable, we shall obtain 
a predicative function of x if we assign a value to <f> 1 2. Thus, to take the 
simplest possible case, if /! (<f> ! z, x) is <f> ! #,the assignment of a value to <f> gives 
us a predicative function of x, in virtue of the definition of " <f> ! x. " But if 
f\ (<f> 1 2, x) contains a first-order function as apparent variable, the assignment 
of a value to <£ ! z gives us a second-order function of x. 

In the third place, we have second-order functions of individuals. These 
will all be derived from functions of the form/! (</> ! z, x) by turning <f> into an 
apparent variable. We do not, therefore, need a new notation for them. 

We have also second-order functions of two first-order functions, or of two 
such functions and an individual, and so on. 

We may now proceed in exactly the same way to third-order matrices, 
which will be functions containing second-order functions as arguments, and 
containing no apparent variables, and no arguments except individuals and 
first-order functions and second-order functions. Thence we shall proceed, as 
before, to third-order functions; and so we can proceed indefinitely. If the 
highest order of variable occurring in a function, whether as argument or as 
apparent variable, is a function of the nth order, then the function in which 
it occurs is of the n + 1th order. We do not arrive at functions of an infinite 
order, because the number of arguments and of apparent variables in a function 
must tye finite, and therefore every function must be of a finite order. Since 
the orders of functions are only defined step by step, there can be no process 
of " proceeding to the limit," and functions of an infinite order cannot occur. 

We will define a function of one variable as predicative when it is of the 
next order above that of its argument,, i.e. of the lowest order compatible with 
its having that argument. If a function has several arguments, and the highest 
order of function occurring among the arguments is the nth, we call the function 
predicative if it is of the n -f- 1th order, i.e. again, if it is of the lowest order 
compatible with its having the arguments it has. A function of several 
arguments is predicative if there is one of its arguments such that, when the 
other arguments have values assigned to them, we obtain a predicative function 
of the one undetermined argument. 

It is important to observe that all possible functions in the above hierarchy 
can be obtained by means of predicative functions and apparent variables. Thus, 
as we saw, second-order functions of an individual x are of the form 

(<f>) ./! (<f> ! z,x) or (gtf>) . /! (<f> ! % x) or (<f>, f) ./! (<f> ! z, -f ! % x) or etc., 
where f is a second-order predicative function. And speaking generally, a 



54 INTRODUCTION [CHAP. 

non-predicative function of the nth order is obtained from a predicative function 
of the nth order by turning all the arguments of the n — 1th order into apparent 
variables. (Other arguments also may be turned into apparent variables.) Thus 
we need not introduce as variables any functions except predicative functions. 
Moreover, to obtain any function of one variable x, we need not go beyond 
predicative functions of two variables. For the function (yjr) ./! (<£> ! 2, ty ! / z, x), 
where / is given, is a function of <j> 1 1z and x, and is predicative. Thus it is of 
the form F ! (cf> I z, x), and therefore (<f>, ty) ./! (<f> ! £ , yfr I z, x) is of the form 
(<£) . F ! (</> ! £, x). Thus speaking generally, by a succession of steps we find that, 
if $ ! u is a predicative function of a sufficiently high order, any assigned non- 
predicative function of x will be of one of the two forms 

where F is a predicative function of <f> I u and x. 

The nature of the above hierarchy of functions may be restated as follows. 
A function, as we saw at an earlier stage, presupposes as part of its meaning 
the totality of its values, or, what comes to the same thing, the totality of 
its possible arguments. The arguments to a function may be functions or 
propositions or individuals. (It will be remembered that individuals were 
defined as whatever is neither a proposition nor a function.) For the present 
we neglect the case in which the argument to a function is a proposition. 
Consider a function whose argument is an individual. This function pre- 
supposes the totality of individuals; but unless it contains functions as 
apparent variables, it does not presuppose any totality of functions. If, 
however, it does contain a function as apparent variable, then it cannot 
be defined until some totality of functions has been defined. It follows that 
we must first define the totality of those functions that have individuals 
as arguments and contain no functions as apparent variables. These are 
the predicative functions of individuals. Generally, a predicative function 
of a variable argument is one which involves no totality except that of 
the possible values of the argument, and those that are presupposed by any 
one of the possible arguments. Thus a predicative function of a variable 
argument is any function which can be specified without introducing new 
kinds of variables not necessarily presupposed by the variable which: is the 
argument. 

A closely analogous treatment can be developed for propositions. Pro- 
positions which contain no functions and no apparent variables may be called 
elementary propositions. Propositions which are not elementary, which contain 
no functions, and no apparent variables except individuals, may be called 
first-order propositions. (It should be observed that no variables except 
apparent variables can occur in a proposition, since whatever contains a real 
variable is a function, not a proposition.) Thus elementary and first-order 
propositions will be values of first-order functions. (It should be remembered 



n] THE AXIOM OF REDUCIBILITY 55 

that a function is not a constituent in one of its values : thus for example 

the function " £ is human " is not a constituent of the proposition' " Socrates 

is human.") Elementary and first-order propositions presuppose no totality 

except (at most) the totality of individuals. They are of one or other of the 

three forms , , , x , , . x , . 

<j>lx; (x).<f>lx; {Qx).<p\x, 

where <f> I x is a predicative function of an individual. If follows that, if p 
represents a variable elementary proposition or a variable first-order propo- 
sition, a function^ is either/(<£ ! x) orf{(x) .<f>lx] or/{(g#) .<f>lx}. Thus 
a function of an elementary or a first-order proposition may always be reduced 
to a function of a first-order function. It follows that a proposition involving 
the totality .of first-order propositions may be reduced to one involving the 
totality of first-order functions ; and this obviously applies equally to higher 
orders. The propositional hierarchy can, therefore, be derived from the 
functional hierarchy, and we may define a proposition of the nth order as 
one which involves an apparent variable of the n — 1th order in the functional 
hierarchy. The propositional hierarchy is never required in practice, and is 
only relevant for the solution of paradoxes ; hence it is unnecessary to go into 
further detail as to the types of propositions. 

VI. The Axiom of Reductibility. 
It remains to consider the " axiom of reducibility." It will be seen that, 
according to the above hierarchy, no statement can be made significantly 
about " all a-functions," where a is some given object. Thus such a notion 
as " all properties of a," meaning " all functions which are true with the 
argument a" will be illegitimate. We shall have to distinguish the order 
of function concerned. We can speak of " all predicative properties of a/**" all 
second-order properties of a," and so on. (If a is not an individual, but an 
object of order w, " second-order properties of a" will mean " functions of 
order n + 2 satisfied by a") But we cannot speak of " all properties of a." 
In some cases, we can see that some statement will hold of " all wth-order 
properties of a," whatever value n may have. In such cases, no practical 
harm results from regarding the statement as being about " all properties of 
a" provided we remember that it is really a number of statements, and not 
a single statement which could be regarded as assigning another property to 
a, over and above all properties. Such cases will always involve some syste- 
matic ambiguity, such as that involved in the meaning of the word "truth," 
as explained above. Owing to this systematic ambiguity, it will be possible, 
sometimes, to combine into a single verbal statement what are really a number 
of different statements, corresponding to different orders in the hierarchy. 
This is illustrated in the case of the liar, where the statement "all A's 
statements are false " should be broken up into different statements referring 
to his statements of various orders, and attributing to each the appropriate 
kind of falsehood. 



56 INTRODUCTION [CHAP. 

The axiom of reducibility is introduced in order to legitimate a great 
mass of reasoning, in which, prima facie, we are concerned with such notions 
as "all properties of a" or "all a-functions," and in which, nevertheless, it 
seems scarcely possible to suspect any substantial error. In order to state 
the axiom, we must first define what is meant by " formal equivalence." Two 
functions </>&, -*}/£ are said to be "formally equivalent" when, with every possible 
argument x, <f>x is equivalent to tyx, i.e. <j>x and yjrx are either both true or 
both false. Thus two functions are formally equivalent when they are satisfied 
by the same set of arguments. The axiom of reducibility is the assumption 
that, given any function <f)fc, there is a formally equivalent predicative function, 
i.e. there is a predicative function which is true when <j>x is true and false 
when <f>x is false. In symbols, the axiom is : 

h : (^) : (f>x . = x . yjrl x. 
For two variables, we require a similar axiom, namely: Given any function 
<j> (&, p), there is a formally equivalent predicative function, i.e. 
\- : (a^r) : <f> (x, y) . = x>y . yfr I (x, y). 

In order to explain the purposes of the axiom of reducibility, and the nature 
of the grounds for supposing it true, we shall first illustrate it by applying it 
to some particular cases. 

If we call a predicate of an object a predicative function which is true of 
that object, then the predicates of an object are only some among its properties. 
Take for example such a proposition as " Napoleon had all the qualities that 
make a great general." We may interpret this as meaning "Napoleon had all 
the predicates that make a great general." Here there is a predicate which is 
an apparent variable. If we put "f(<j> I z)" for "<f> I % is a predicate required 
in a great general," our proposition is 

(<£) :/(</> ! z) implies <j> ! (Napoleon). 
Since this refers to a totality of predicates, it is not itself a predicate of 
Napoleon. It by no means follows, however, that there is not some one predicate 
common and peculiar to great generals. In fact, it is certain that there is such 
a predicate. For the number of great generals is finite, and each of them 
certainly possessed some predicate not possessed by any other human being 
—for example, the exact instant of his birth. The disjunction of such predicates 
will constitute a predicate common and peculiar to great generals*. If we 
call this predicate yfr ! z, the statement we made about Napoleon was equi- 
valent to yfr I (Napoleon). And this equivalence holds equally if we substitute 
any other individual for Napoleon. Thus we have arrived at a predicate which 
is always equivalent to the property we ascribed to Napoleon, i.e. it belongs 
to those objects which have this property, and to no others. The axiom of 
reducibility states that such a predicate always exists, i.e. that any property 

* When a (finite) set of predicates is given by actual enumeration, their disjunction is a 
predicate, because no predicate occurs as apparent variable in the disjunction. 



It] THE AXIOM OP REDUCTIBILITY 57 

of an object belongs to the same collection of objects as those that possess 
some predicate. 

We may next illustrate our principle by its application to identity. In 
this connection, it has a certain affinity with Leibniz's identity of indiscernibles. 
It is plain that, if x and y are identical, and <f>cc is true, then $y is true. Here 
it cannot matter what sort of function $5b may be : the statement must hold 
for any function. But we cannot say, conversely : " If, with all values of <£, 
<j>x implies <j>y, then x and y are identical " ; because "all values of <f> " is 
inadmissible. If we wish to speak of "all values of <f>," we must confine 
ourselves to functions of one order. We may confine <f> to predicates, or to 
second-order functions, or to functions of any order we please. But we must 
necessarily leave out functions of all but one order. Thus we shall obtain, so 
to speak, a hierarchy of different degrees of identity. We may say " all the 
predicates of x belong to y," " all second-order properties of x belong to y," 
and so on. Each of these statements implies all its predecessors : for 
example, if all second-order properties of x belong to y, then all predicates 
of x belong to y, for to have all the predicates of a; is a second-order property, 
and this property belongs to x. But we cannot, without the help of an axiom, 
argue conversely that if all the predicates of # belong to y, all the second-order 
properties of x must also belong to y. Thus we cannot, without the help of 
an axiom, be sure that x and y are identical if they have the same predicates. 
Leibniz's identity of indiscernibles supplied this axiom. It should be observed 
that by " indiscernibles " he cannot have meant two objects which agree as to 
all their properties, for one of the properties of x is to be identical with x, 
and therefore this property would necessarily belong to y if x and y agreed 
in all their properties. Some limitation of the common properties necessary 
to make things indiscernible is therefore implied by the necessity of an axiom. 
For purposes of illustration (not of interpreting Leibniz) we may suppose the 
common properties required for indiscernibility to be limited to predicates. 
Then the identity of indiscernibles will state that if x and y agree as to 
all their predicates, they are identical. This can be proved if we assume the 
axiom of reducibility. For, in that case, every property belongs to the same 
collection of objects as is defined by some predicate. Hence there is some 
predicate common and peculiar to the objects which are identical with x. 
This predicate belongs to x, since x is identical with itself; hence it belongs 
to y, since y has all the predicates of x ; hence y is identical with x. It 
follows that we may define x and y as identical when all the predicates of x 
belong to y, i.e. when (<f>) : <f> ! x . D . <f> ! y. We therefore adopt the following 
definition of identity*: 

x=y. = :(<f>):<f>lx.D.<f>ly Df. 

* Note that in this definition the second sign of equality is to be regarded as combining with 
"T>( " to form one symbol; what is defined is the sign of equality not followed by the letters "Df." 



58 INTRODUCTION [CHAP. 

But apart from the axiom of reducibility, or some axiom equivalent in this 
connection, we should be compelled to regard identity as indefinable, and to 
admit (what seems impossible) that two objects may agree in all their pre- 
dicates without being identical. 

The axiom of reducibility is even more essential in the theory of classes. 
It should be observed, in the first place, that if we assume the existence of 
classes, the axiom of reducibility can be proved. For in that case, given any 
function $z of whatever order, there is a class a consisting of just those 
objects which satisfy $z. Hence "<£#" is equivalent to "x belongs to or." 
But " x belongs to a " is a statement containing no apparent variable, and is 
therefore a predicative function of x. Hence if we assume the existence of 
classes, the axiom of reducibility becomes unnecessary. The assumption of 
the axiom of reducibility is therefore a smaller assumption than the assump- 
tion that there are classes. This latter assumption has hitherto been made 
unhesitatingly. However, both on the ground of the contradictions, which 
require a more complicated treatment if classes are assumed, and on the ground 
that it is always well to make the smallest assumption required for proving 
our theorems, we prefer to assume the axiom of reducibility rather than the 
existence of classes. But in order to explain the use of the axiom in dealing 
with classes, it is necessary first to explain the theory of classes, which is a 
topic belonging to Chapter III. We therefore postpone to that Chapter the 
explanation of the use of our axiom in dealing with classes. 

It is worth while to note that all the purposes served by the axiom of 
reducibility are equally well served if we assume that there is always a function 
of the nth order (where n is fixed) which is formally equivalent to <f>x, what- 
ever may be the order of <£&. Here we shall mean by "a function of the nth 
order" a function of the nth order relative to the arguments to <££ ; thus if 
these arguments are absolutely of the rath order, we assume the existence of 
a function formally equivalent to <f>& whose absolute order is the m + nth. The 
axiom of reducibility in the form assumed above takes n = 1, but this is not 
necessary to the use of the axiom. It is also unnecessary that n should be the 
same for different values of ra; what is necessary is that n should be constant 
so long as m is constant. What is needed is that, where extensional functions 
of functions are concerned, we should be able to deal with any a-function by 
means of some formally equivalent function of a given type, so as to be able 
to obtain results which would otherwise require the illegitimate notion of 
" all a-functions " ; but it does not matter what the given type is. It does 
not appear, however, that the axiom of reducibility is rendered appreciably 
more plausible by being put in the above more general but more complicated 
form. 

The axiom of reducibility is equivalent to the assumption that "any 



II] THE AXIOM OF REDUCIBILITY 59 

combination or disjunction of predicates* is equivalent to a single predicate," 
i.e. to the assumption that, if we assert that x has all the predicates that 
satisfy a function f {<f> I z), there is some one predicate which x will have 
whenever our assertion is true, and will not have whenever it is false, and 
similarly if we assert that x has some one of the predicates that satisfy a function 
f(<f>lz). For by means of this assumption, the order of a non-predicative function 
can be lowered by one; hence, after some finite number of steps, we shall be able 
to get from any non-predicative function to a formally equivalent predicative 
function. It does not seem probable that the above assumption could be 
substituted for the axiom of reducibility in symbolic deductions, since its use 
would require the explicit introduction of the further assumption that by a 
finite number of downward steps we can pass from any function to a predicative 
function, and this assumption -could not well be made without developments 
that are scarcely possible at an early stage. But on the above grounds it seems 
plain that in fact, if the above alternative axiom is true, so is the axiom of 
reducibility. The converse, which completes the proof of equivalence, is of 
course evident. 

VII. Reasons for Accepting the Axiom of Reducibility. 

That the axiom of reducibility is self-evident is a proposition which can 
hardly be maintained. But in fact self-evidence is never more than a part of 
the reason for accepting an axiom, and is never indispensable. The reason 
for accepting an axiom, as for accepting any other proposition, is always 
largely inductive, namely that many propositions which are nearly indubitable 
can be deduced from it, and that no equally plausible way is known by which 
these propositions could be true if the axiom were false, and nothing which is 
probably false can be deduced from it. If the axiom is apparently self-evident, 
that only means, practically, that it is nearly indubitable; for things have 
been thought to be self-evident and have yet turned out to be false. And if 
the axiom itself is nearly indubitable, that merely adds to the inductive 
evidence derived from the fact that its consequences are nearly indubitable : 
it does not provide new evidence of a radically different kind. Infallibility is 
never attainable, and therefore some element of doubt should always attach 
to every axiom and to all its consequences. In formal logic, the element of 
doubt is less than in most sciences, but it is not absent, as appears from the 
fact that the "paradoxes followed from premisses which were not previously 
known to require limitations. In the case of the axiom of reducibility, the 
inductive evidence in its favour is very strong, since the reasonings which it 
permits and the results to which it leads are all such as appear valid. But 
although it seems very improbable that the axiom should turn out to be false, 

* Here the combination or disjunction is supposed to be given intensionally. If given exten- 
sionally (i.e. by enumeration), no assumption is required ; but in this case the number of 
predicates concerned must be finite. 



60 INTRODUCTION [CHAP. 

it is by no means improbable that it should be found to be deducibie from 
some other more fundamental and more evident axiom. It is possible that the 
use of the vicious-circle principle, as embodied in the above hierarchy of types, 
is more drastic than it need be, and that by a less drastic use the necessity 
for the axiom might be avoided. Such changes, however, would not render 
anything false which had been asserted on the basis of the principles explained 
above : they would merely provide easier proofs of the same theorems. There 
would seem, therefore, to be but the slenderest ground for fearing that the 
use of the axiom of reducibility may lead us into error. 

VIII. The Contradictions. 
We are now in a position to show how the theory of types affects the 
solution of the contradictions which have beset mathematical logic. For this 
purpose, we shall begin by an enumeration of some of the more important and 
illustrative of these contradictions, and shall then show how they all embody 
vicious-circle fallacies, and are therefore all avoided by the theory of types. It 
will be noticed that these paradoxes do not relate exclusively to the ideas of 
number and quantity. Accordingly no solution can be adequate which seeks 
to explain them merely as the result of some illegitimate use of these ideas. 
The solution must be sought in some such scrutiny of fundamental logical 
ideas as has been attempted in the foregoing pages. 

(1) The oldest contradiction of the kind in question is the Epimenides. 
Epimenides the Cretan said that all Cretans were liars, and all other state- 
ments made by Cretans were certainly lies. Was this a lie ? The simplest form 
of this contradiction is afforded by the man who says "I am lying"; if he is 
lying, he is speaking the truth, and vice versa. 

(2) Let w be the class of all those classes which are not members of 
themselves. Then, whatever class x may be, "« is a w" is equivalent to "oc is 
not an w." Hence, giving to w the value w, "w is a w" is equivalent to 
"w is not a w." 

(3) Let T be the relation which subsists between two relations R and 8 
whenever R does not have the relation R to S. Then, whatever relations 
R and S may be, "R has the relation T to S" is equivalent to "R does not 
have the relation R to S." Hence, giving the value T to both R and 8, 
"T has the relation T to T" is equivalent to "T does not have the relation 
T to T." 

(4) Burali-Forti's contradiction* may be stated as follows : It can be 
shown that every well-ordered series has an ordinal number, that the series of 
ordinals up to and including any given ordinal exceeds the given ordinal by 
one, and (on certain very natural assumptions) that the series of all ordinals 
(in order of magnitude) is well-ordered. It follows that the series of .all 

* "Una questione sui numeri transfiniti," Rendiconti del circolo matematico di Palermo, Vol. 
xi. (1897). See *256. 



n] ENUMERATION OF CONTRADICTIONS 61 

ordinals has an ordinal number, X2 say. But in that case the series of all 
ordinals including X2 has the ordinal number O + 1, which must be greater 
than .XI. Hence XI is not the ordinal number of all ordinals. 

(5) The number of syllables in the English names of finite integers 
tends to increase as the integers grow larger, and must gradually increase 
indefinitely, since only a finite number of names can be made with a given 
finite number of syllables. Hence the names of some integers must consist of 
at least nineteen syllables, and among these there must be a least. Hence "the 
least integer not nameable in fewer than nineteen syllables" must denote a 
definite integer; in fact, it denotes 111,777. But "the least integer not 
nameable in fewer than nineteen syllables " is itself a name consisting of 
eighteen syllables; hence the least integer Ikot nameable in fewer than nine- 
teen syllables can be named in eighteen syllables, which is a contradiction*. 

(6) Among transfinite ordinals some can be defined, while others can not; 
for the total number of possible definitions is K f, while the number of trans- 
finite ordinals exceeds K . Hence there must be indefinable ordinals, and 
among these there must be a least. But this is defined as " the least indefinable 
ordinal," which is a contradiction J. 

(7) Richard's paradox § is akin to that of the least indefinable ordinal. It 
is as follows : Consider all decimals that can be defined by means of a finite 
number of words ; let E be the class of such decimals. Then E has tf terms; 
hence its members can be ordered as the 1st, 2nd, 3rd, .... Let N be a number 
defined as follows: If the nth figure in the nth decimal is p, let the nth 
figure in N be p + 1 (or 0, if p = 9). Then N is different from all the members 
of E, since, whatever finite value n may have, the nth figure in N is different 
from the nth. figure in the nth of the decimals composing E, and therefore N 
is different from the nth decimal. Nevertheless we have defined N in a finite 
number of words, and therefore N ought to be a member of E. Thus N both 
is and is not a member of E. 

In all the above contradictions (which are merely selections from an 
indefinite number) there is a common characteristic, which we may describe 
as self-reference or reflexiveness. The remark of Epimenides must include 
itself in its own scope. If all classes, provided they are not members of them- 
selves, are members of w, this must also apply to w ; and similarly for the 

* This contradiction was suggested to us by Mr G. G. Berry of the Bodleian Library. 

■f N is the number of finite integers. See *123. 

X Cf. Konig, "Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem," Math. 
Annalen, Vol. lxi. (1905); A. C. Dixon, "On 'well-ordered' aggregates," Proc. London Math. 
Soc, Series 2, Vol. iv. Part i. (1906); and E. W. Hobson, "On the Arithmetic Continuum," ibid. 
The solution offered in the last of these papers depends upon the variation of the "apparatus of 
definition," and is thus in outline in agreement with the solution adopted here. But it does not 
invalidate the statement in the text, if "definition" is given a constant meaning. 

§ Cf. Poincare, "Les mathematiques et la logique," Revue de Metaphysiqve et de Morale, 
Mai 1906, especially sections vn. and ix.; also Peano, Revista de Mathematiea, Vol. vm. No. 5 
(1906), p. 149 fit. ■ 



62 INTRODUCTION [OHAP. 

analogous relational contradiction. In the cases of names and definitions, the 
paradoxes result from considering non-nameability and indefinability as ele- 
ments in names and definitions. In the case of Burali-Forti's paradox, the 
series whose ordinal number causes the difficulty is the series of all ordinal 
numbers. In each contradiction something is said about all cases of some kind, 
and from what is said a new case seems to be generated, which both is and is not 
of the same kind as- the cases of which all were concerned in what was said. 
But this is the characteristic of illegitimate totalities, as we defined them in 
stating the vicious-circle principle. Hence all our contradictions are illustra- 
tions of vicious-circle fallacies. It only remains to show, therefore, that the 
illegitimate totalities involved are excluded by the hierarchy of types which 
we have constructed. 

(1) When a man says "I am lying," we may interpret his statement as: 
"There is a proposition which I am affirming and which is false." That is to 
say, he is asserting the truth of -some value of the function "I assert p, and p 
is false." But we saw that the word "false" is ambiguous, and that, in order 
to make it unambiguous, we must specify the order of falsehood, or, what comes 
to the same thing, the order of the proposition to which falsehood is ascribed. 
We saw also that, if p is a proposition of the nth order, a proposition in which 
p occurs as an apparent variable is not of the nth order, but of a higher order. 
Hence the kind of truth or falsehood which can belong to the statement "there 
is a proposition p which I am affirming and which has falsehood of the nth 
order" is truth or falsehood of a higher order than the nth. Hence the state- 
ment of Epimenides does not fall within its own scope, and therefore no 
contradiction emerges. 

If we regard the-statement "I am lying " as a compact way of simultaneously 
making all the following statements: te I am asserting a false proposition of the 
first order," "I am asserting a false proposition of the second order," and so on, 
we find the following curious state of things: As no proposition of the first 
order is being asserted, the statement "I am asserting a false proposition of 
the first order" is False. This statement is of the second order/hence the 
statement "I am making a false statement Of the second order" is true. This 
is a statement of the third order, -and is the only statement of the third order 
which is being made. Hence the statement "I am making a false statement 
of the third order" is false. Thus we see that the statement "I am making a 
false statement of order 2n + 1" is false, while the statement "I am making 
a false statement of order 2n" is true. But in this state of things there is no 
contradiction. 

(2) In order to solve the contradiction about the class of classes which are 
not members of themselves, we shall assume, what will be explained in the 
next Chapter, that a proposition about a class is always to be reduced to a 
statement about a function which defines the class, i.e. about a function which 



II] VICIOUS-CIRCLE FALLACIES 63 

is satisfied by the members of the class and by no other arguments. Thus a 
class is an object derived from a function and presupposing the function, just 
as, for example, (x) . <j>x presupposes the function <p£. Hence a class cannot, 
by the vicious-circle principle, significantly be the argument to its defining 
function, that is to say, if we denote by u z(<f)z)" the class defined by $%, the 
symbol "<£ [z (fa)}" must be meaningless. Hence a class neither satisfies nor 
does not satisfy its defining function, and therefore (as will appear more fully 
in Chapter III) is neither a member of itself nor not a member of itself. This 
is an immediate consequence of the limitation to the possible arguments to a 
function which was explained at the beginning of the present Chapter. Thus 
if a is a class, the statement "a is not a member of a" is always meaningless, 
and there is therefore ho sense in the phrase "the class of those classes which 
are not members of themselves." Hence the contradiction which results from 
supposing that there is such a class disappears. 

(3) Exactly similar remarks apply to "the relation which holds between 
R and S whenever R does not have the relation R to S." Suppose the 
relation R is defined by a function <j> (#, y), i.e. R holds between x and y 
whenever <f> (x, y) is true, but not otherwise. Then in order to interpret 
"R has the relation R to S," we shall have to suppose that R and S can 
significantly be the arguments to 4>. But (assuming, as will appear in 
Chapter HI, that R presupposes its defining function) this would require 
that </> should be able to take as argument , an object which is defined in 
terms of ^>, and this no function can do, as we saw at the beginning of this 
Chapter. Hence "R has the relation R to S" is meaningless, and the contra- 
diction ceases. 

(4) The solution of Burali-FortTs contradiction requires some further 
developments for its solution. At this stage, it must suffice to observe that 
a series is a relation, and an ordinal number is a class of series. (These state- 
ments are justified in the body of the-work.) Hence a series of ordinal numbers 
is a relation between classes of relations, and is of higher type than any of the 
series which are members of the ordinal numbers in question. Burali-Forti's 
"ordinal number of all ordinals" must be the ordinal number of all ordinals of 
a given type, and must therefore be of higher type than any of these ordinals. 
Hence it is not one of these ordinals, and there is no contradiction in its being 
greater than any of them *. 

(5) The paradox about "the least integer not nameabJe in fewer than 
nineteen syllables" embodies, as is at once obvious, a vicious-circle fallacy. 
For the word " nameable" refers to the totality of names, and yet is allowed 
to occur in what professes to be one among names. Hence there can be no 
such thing as a totality of names, in the sense in which the paradox speaks 

* The solution of Burali-Forti's paradox by means of the theory of types is given in detail ia 
*256. 



64 INTRODUCTION [CHAP. 

of "names." It is easy to see that, in virtue of the hierarchy of functions; 
the theory of types renders a totality of "names" impossible. We may, in 
fact, distinguish names of different orders as follows: (a) Elementary names 
will be such as are true "proper names," i.e. conventional appellations not 
involving any description, (b) First-order names will be such as involve a 
description by means of a first-order function; that is to say, if <j> I & is a first- 
order function, "the term which satisfies <j>\£" will be a first-order name, 
though there will not always be an object named by this name, (c) Second- 
order names will be such as involve a description by means of a second-order 
function; among such names will be those involving a reference to the totality 
of first-order names. And so we can proceed through a whole hierarchy. But 
at no stage can we give a meaning to the word "nameable" unless we specify 
the order of names to be employed; and any name in which the phrase "name- 
able by names of order n " occurs is necessarily of a higher order than the nth. 
Thus the paradox disappears. 

The solutions of the paradox about the least indefinable ordinal and 
of Richard's paradox are closely analogous to the above. The notion of 
"definable," which occurs in both, is nearly the same as "nameable," which 
occurs in our fifth paradox: "definable" is what "nameable" becomes 
when elementary names are excluded, i.e. "definable" means "nameable by 
a name which is not elementary." But here there is the same ambiguity 
as to type as there was before, and the same need for the addition of words 
which specify the type to which the definition is to belong. And however 
the type may be specified, "the least ordinal not definable by definitions of 
this type" is a definition of a higher type; and in Richard's paradox, when 
we confine ourselves, as we must, to decimals that have a definition of a given 
type, the number N, which causes the paradox, is found to have a definition 
which belongs to a higher type, and thus not to come within the scope of our 
previous definitions. 

An indefinite number of other contradictions, of similar nature to the 
above seven, can easily be manufactured. In all of them, the solution is 
of the same kind. ,In all of them, the appearance of contradiction is pro- 
duced by the presence of some word which has systematic ambiguity of 
type, such as truth, falsehood, function, property, class, relation, cardinal, 
ordinal, name, definition. Any such word, if its typical ambiguity is over- 
looked, will apparently generate a totality containing members defined in 
terms of itself; and will thus give rise to vicious-circle fallacies. In most 
cases, the conclusions of arguments which involve vicious-circle fallacies 
will not be self-contradictory, but wherever we have an illegitimate totality, 
a little ingenuity will enable us to construct a vicious-circle fallacy leading 
to a contradiction, which disappears as soon as the typically ambiguous words 
are rendered typically definite, i.e. are determined as belonging to this or that 
type. 



Il] VICIOUS-CIRCLE FALLACIES 65 

Thus the appearance of contradiction is always due to the presence of words 
embodying a concealed typical ambiguity, and the solution of the apparent 
contradiction lies in bringing the concealed ambiguity to light. 

In spite of the contradictions which result from unnoticed typical 
ambiguity, it is not desirable to avoid words and symbols which have 
typical ambiguity. Such words and symbols embrace practically all the 
ideas with which mathematics and mathematical logic are concerned: the 
systematic ambiguity is the result of a systematic analogy. That is to say, in 
almost all the reasonings which constitute mathematics and mathematical 
logic, we are using ideas which may receive any one of an infinite number of 
different typical determinations, any one of which leaves the reasoning valid. 
Thus by employing typically ambiguous words and symbols, we are able to make 
one chain of reasoning applicable to any one of an infinite number of different 
cases, which would not be possible if we were to forego the use of typically 
ambiguous words and symbols. 

Among propositions wholly expressed in terms of typically ambiguous 
notions practically the only ones which may differ, in respect of truth or false- 
hood, according to the typical determination which they receive, are existence- 
theorems. If we assume that the total number of individuals is n/then the 
total number of classes of individuals is 2 n , the total number of classes of classes 
of individuals is 2 2 , and so on. Here n may be either finite or infinite, and in 
either case 2 n > n. Thus cardinals greater than n but not greater than 2 n exist 
as applied to classes of classes, but not as applied to classes of individuals, so 
that whatever may be supposed to be the number of individuals, there will be 
existence-theorems which, hold for higher types but not for lower types. Even 
here, however, so long as the number of individuals is not asserted, but is 
merely assumed hypothetical^, we may replace the type of individuals by any 
other type, provided we make a corresponding change in all the other types 
occurring in the same context. That is, we may give the name "relative in- 
dividuals" to the members of an arbitrarily chosen type t, and the name 
"relative classes of individuals" to classes of "relative individuals," and so on. 
Thus so long as only hypothetical are concerned, in which existence-theorems 
for one type are shown to be implied by existence-theorems for another, only 
relative types are relevant even in existence-theorems. This applies also to cases 
where the hypothesis (and therefore the conclusion) is asserted, provided the 
assertion holds for any type, however chosen. For example, any type has at 
least one member; hence any type which consists of classes, of whatever order, 
has at least two members. But the further pursuit of these topics must be left 
to the body of the work. 



R&W I 



CHAPTER III 

INCOMPLETE SYMBOLS 

(1) Descriptions.. By an "incomplete" symbol we mean a symbol which 
is not supposed to have any meaning in isolation, but is only defined in 

d f b 

'certain contexts. In ordinary mathematics, for example, ^ and J^ are in- 
complete symbols: something has to be supplied before we have anything 
significant. Such symbols have what may be called a "definition in use." 

Thus if we put 

2 2 3* & 

V! = £_+-£- + ;?- Df, 

y da? df dz* 
we define the use of V 2 , but V 2 by itself remains without meaning. This dis- 
tinguishes such symbols from what (in a generalized sense) we may call proper 
names: "Socrates," for example, stands for a certain man, and therefore has 
a meaning by itself, without the need of any context. If we supply a context, 
as in "Socrates is mortal," these words express a fact of which Socrates him- 
self is a constituent: there is a certain object, namely Socrates, which does 
have the property of mortality, and this object is a constituent of the complex 
fact which we assert when we say "Socrates is mortal." But in other cases, 
this simple analysis fails us. Suppose we say: "The round square does not 
exist." It seems plain that this is a true proposition, yet we cannot regard it 
as denying the existence of a certain object called " the round square." For 
if there were such an object, it would exist: we cannot first assume that there 
is a certain object, and then proceed to deny that there is such an object. 
Whenever the grammatical subject of a proposition can be supposed not to 
exist without rendering the proposition meaningless, it is plain that the 
grammatical subject is not a proper name, i.e. not a name directly representing 
some object. Thus in all such cases, the proposition must be capable of being 
so analysed that what was the grammatical subject shall have disappeared. 
Thus when we say " the round square does not exist," we may, as a first 
attempt at such analysis, substitute " it is false that there is an object as which 
is both round and square." Generally, when "the so-and-so" is said not to 
exist, we have a proposition of the form* 

"~E l(ix)(<f>x)" 
i.e. ~{(ac):^c.=*.0 = c}, 

or some equivalent. Here the apparent grammatical subject (ix)(<f>x) has 
completely, disappeared; thus in "~E!(?#)(<M," (»*)(**) is an ^complete 

symbol. 

* Cf. pp. 30, 31. 



CHAP. Ill] DESCRIPTIONS 67 

By an extension of the above argument, it can easily be shown that 
(ix) (<f>x) is always an incomplete symbol. Take, for example, the following 
proposition: "Scott is the author of Waverley." [Here "the author of 
Waverley" is (ix) (x wrote Waverley).] This proposition expresses an identity; 
thus if " the author of Waverley " could be taken as a proper name, and sup- 
posed to stand for some object c, the proposition would be " Scott is c." But 
if c is any one except Scott, this proposition is false; while if c is Scott, the 
proposition is "Scott is Scott," which is trivial, and plainly different from 
" Scott is the author of Waverley." Generalizing, we see that the proposition 

a = (ix)(<f>x) 
is one which may be true or may be false, but is never merely trivial, like 
a = a; whereas, if (ix) ($x) were a proper name, a = {ix) (<f>x) would necessarily 
be either false or the same as the trivial proposition a = a. We may express 
this by saying that a = (ix)(<f>x) is not a value of the propositional function 
a = y, from which it follows that (ix) (<f>x) is not a value of y. But since y 
may be anything, it follows that (ix) (<f>x) is nothing. Hence, since in use it 
has meaning, it must be an incomplete symbol. 

It might be suggested that " Scott is the author of Waverley " asserts that 
"Scott" and "the author of Waverley" are two names for the same object. 
But a little reflection will show that this would be a mistake. For if that 
were the meaning of " Scott is the author of Waverley," what would be required 
for its truth would be that Scott should have been called the author of 
Waverley: if he had been so called, the proposition would be true, even if 
some one else had written Waverley; while if no one called him so, the pro- 
position-would be false, even if he had written Waverley. But in fact he was 
the author of Waverley at a time when no one called him so, and he would 
not have been the author if every one had called him so but some one else 
had written Waverley. Thus the proposition "Scott is the author of Waverley" 
is not a proposition about names, like "Napoleon is Bonaparte"; and this 
illustrates the sense in which "the author of Waverley " differs from a true 
proper name. 

Thus all phrases (other than propositions) containing the word the (in the 
singular) are incomplete symbols: they have a meaning in use, but not in 
isolation. For " the author of Waverley " cannot mean the same as " Scott," 
or " Scott is the author of Waverley " would mean the same as " Scott is 
Scott," which it plainly does not; nor can "the author of Waverley" mean 
anything other than " Scott," or " Scott is the author of Waverley " would be 
false. Hence "the author of Waverley" means nothing. 

It follows from the above that we must not attempt to define " {ix) (<£#)," 
but must define the uses of this symbol, i.e. the propositions in whose symbolic 
expression it occurs. Now in seeking to define the uses of this symbol, it is 
important to observe the import of propositions in which it occurs. Take as 

5—2 



68 INTRODUCTION [CHAP. 

an illustration: "The author of Waverley was a poet." This implies (1) that 
Waverley was written, (2) that it was written by one man, and not in collabora- 
tion, (3) that the one man who wrote it was a poet. If any one of these fails, 
the proposition is false. Thus " the author of « Slawkenburgius on Noses ' was 
a poet " is false, because no such book was ever written; " the author of ' The 
Maid's Tragedy' was a poet" is false, because this play was written by 
Beaumont and Fletcher jointly. These two possibilities of falsehood do not 
arise if we say " Scott was a poet." Thus our interpretation of the uses of 
(ix)(<f>x) must be such as to allow for them. Now taking §x to replace 
" x wrote Waverley," it is plain that any statement apparently about (ix) {<bx) 
requires (1) (gar) . (<f>x) and (2) </>x . 4>y . D XtV . x = y; here (1) states that at 
least one object satisfies <f>x, while (2) states that at most one object satisfies 
<f>x. The two together are equivalent to 

(ftc):<f>x.= x .x=c, 
which we defined as E ! (ix) (<f>x). 

Thus "El(ix)(<f>x)" must be part of what is affirmed by any proposition 
about (ix)(<f>x). If our proposition is/ [(ix)(<f>x)}, what is further affirmed is 
fc, if <f>x . = x . x = c. Thus we have 

f{(ix) (4>x)}. = :(Rc):<j>x.= x .x = c:fc Df, 
i.e. "the x satisfying <f>x satisfies /#" is to mean: "There is an object c such 
that <fix is true when, and only when, x is c, and/c is true," or, more exactly: 
" There is a c such that '<j>x' is always equivalent to ( x is c,' and/c." In this, 
"(w) (<px)" has completely disappeared; thus "(ix)(<f>x)" is merely symbolic, 
and does not directly represent an object, as single small Latin letters are 
assumed to do*. 

The proposition " a = (ix)(<f>x)" is easily shown to be equivalent to 
"<f>x . = x ,x=a." For, by the definition, it is 

(gc) z<j)x.= x .x=c:a = c } 
i.e. " there is a c for which <f>x.= x . x = c, and this c is a," which is equivalent 
to " <px . = x . x = a." Thus " Scott is the author of Waverley " is equivalent to : 

"'x wrote Waverley' is always equivalent to ' x is Scott,'" 
i.e. " x wrote Waverley " is true when x is Scott and false when x is not Scott. 

Thus although "(ix)((f)x)" has no meaning by itself, it may be substituted 
for y in any propositional function/?/, and we get a significant proposition, 
though not a value of fy. 

When /{(?#) (<£#)}, as above defined, forms part of some other proposition, 
we shall say that (ix) (<f)x) has a secondary occurrence. When (ix) (<)>x) has 
a secondary occurrence, a proposition in which it occurs may be true even 
when (ix)(<f>x) does not exist. This applies, e.g. to the proposition: "There 

* We shall generally write "/('*) (0 X )" ra* ner than "/{('*) (<P X )}" in future. 



Ill] THE SCOPE OF A DESCRIPTION 69 

is no such person as the King of France." We may interpret this as 

~{El(ix)(<l>x)}, 
or as <^» {(gc) . c = (ix) (<j>x)}, 

if " <f>x" stands for " x is King of France." In either case, what is asserted is 
that a proposition p in which (ix) (<f>x) occurs is false, and this proposition p 
is thus part of a larger proposition. The same applies to such a proposition 
as the following: " If France were a monarchy, the King of France would be 
of the House of Orleans." 

It should be observed that such a proposition as 

is ambiguous; it may deny f{(ix)(<f>x)}, in which case it will be true if 
(ix) (<fix) does not exist, or it may mean 

(gc) :(f>x.== x .x = c: ~/c, 
in which case it can only be true if (ix) (<f>x) exists. In ordinary language, 
the latter interpretation would usually be adopted. For example, the propo- 
sition " the King of France is not bald " would usually be rejected as false, 
being held to mean "the King of France exists and is not bald," rather than 
"it is false that the King of France exists and is bald." When (ix)((f>x) 
exists, the two interpretations of the ambiguity give equivalent results; but 
when (ix) (<f>x) does not exist, one interpretation is true and one is false. It 
is necessary to be able to distinguish these in our notation; and generally, if 
we have such propositions as 

^(ix)(<f>x).D.p, 

p.^.^(ix)(^>x), 

•>/r (ix) (<f>x) . D . x 0#) (<H> 
and so on, we must be able by our notation to distinguish whether the whole 
or only part of the proposition concerned is to be treated as the "f(ix) (<f>x)" 
of our definition. For this purpose, we will put " [(ix) (<f>x)]" followed by dots 
at the beginning of the part (or whole) which is to be taken as f(ix) (<f>x), the 
dots being sufficiently numerous to bracket off the f(ix)(<f>x); i.e. f(ix)(<f>x) 
is to be everything following the dots until we reach an equal number of dots 
not signifying a logical product, or a greater number signifying a logical pro- 
duct, or the end of the sentence, or the end of a bracket enclosing "[(ix) (</>#)]." 
Thus 

[(ix) (</>#)] ■ "f 0*0 (0*0 ■ D ■ P 
will mean (gc) : <f>x . = x . x = c : -tyc : D . p, 

but [(ix) (<f>x)] : ^ ( ix) (<f>x) . D . p 

will mean (gc) : <j>x . = x . x = c : tyc . D . p. 

It is important to distinguish these two, for if (ix)(<f>x) does not exist, the 
first is true and the second false. Again 

[(ix) (<f>x)] . ~ yjr (ix) ((f>x) 



70 INTRODUCTION [CHAP. 

will mean (gc) : <frx .= x .x = c : ~ yjrc, 

while ~ {[(ix) ((fix)] . \jr (ix) (<f>x)} 

will mean ~ {(3 C ) :<j>x ,= x .x = c: yjrc}. 

Here again, when (7a?) (cfrx) does not exist, the first is false and the second true. 

In order to avoid this ambiguity in propositions containing (ix)(<f>x), we 
amend our definition, or rather our notation, putting 

[(lx)((f>x)],f(ix)((j)x). = :('g L c):<f>x.= x .x=:c:/c Df. 
By means of this definition, we avoid any doubt as to the portion of our 
whole asserted proposition which is to be treated as the "f(ix)(^>x)" of the 
definition. This portion will be called the scope of (ix) (<f>x). Thus in 

[(ix) (<f>x)] .f{ix) (<f>x) . D . p 
the scope of (ix)(<j>x) i$f(ix)(<f>x); but in 

[(ix) (<j>x)] :f(ix) (<)>x) .D.p 
the scope is f(i%) (##)■• ^ -P'> 

in ~ {[(ix) (<f>x)] ./(ix) (4>x)} 

the scope is / (ix) (<f>x); but in 

[(ix)(<t>x)-].~f(ix)(4>x) 
the scope is ~y(?#) (<f>x). 

It will be seen that when (ix) (<px) has the whole of the proposition 
concerned for its scope, the proposition concerned cannot be true unless 
E ! (ix)(<jix); but when' (ix) ($x) has only part of the proposition concerned 
for its scope, it may often be true even when (ix) (cf>x) does not exist. It will 
be seen further that when E ! (ix) (<f>x), we may enlarge or diminish the scope 
of (ix)(if>x) as much as we please without altering the truth-value of any 
proposition in which it occurs. 

If a proposition contains two descriptions, say (ix) (<px) and (ix)(yjrx), 
we have to distinguish which of them has the larger scope, i.e. we have to 
distinguish 

(1) [(ix) (<H] : [(**) (yjrx)].f{(ix) (<f>x), (ix) (fx)}, 

(2) [(ix) (yfrx)] : [(ix) (<f>x)] . f {(ix) (<j>x), (ix) (yjrx)}. 
The first of these, eliminating (ix)(<f>x), becomes 

(3) (gc) : <f>x.= x .x = c: [(ix) (fx)] . f {c, (ix) (fx)}, 
which, eliminating (ix) (yjrx), becomes 

(4) (gc) :.<j>x.= x . x = c:.(<zd):fx.= x .x=c :f(c, d), 

and the same proposition results if, in (1), we eliminate first (ix)(yjrx) and 
then (ix)(<f)x). Similarly (2) becomes, when (ix)(<f>x)< and (ix)(yjrx) are 
eliminated, 

(5) fad) :. yfrx . = x . x = d :. (gc) :<}>x.= x .x = c :/(c, d). 

(4) and (5) are equivalent, so that the truth-value of a proposition contain- 
ing two descriptions is independent of the question which has the larger scope. 



Ill] CLASSES 71 

It will be found that, in most cases in which descriptions occur, their 
scope is, in practice, the smallest proposition enclosed in dots or other brackets 
in which they are contained. Thus for example 

[(**) (**)] ■ 1 0*) (</>*) ■ D ■ [(wO (W] ■ X (w) (W 
will occur much more frequently than 

[(M>) (£a?)] :. -^ ( *»). ( W ■ 3 ■ X O) (**)• 

For this reason it is convenient to decide that, when the scope of an occurrence 

of (ix) (<f>x) is the smallest proposition, enclosed in dots or other brackets, in 

which the occurrence in question is contained, ,the scope need not be indicated 

hy"[(ix)(<f>x)l" Thus e.g. 

p . D . a = (ix) (<£#) 

will mean p . D . [(ix) (4>x)~\ • a = (J x ) ( W '■> 

and p.D. (ga) . a = (?#) (0a;) 

will mean p.D. (g«) . [(?&•) (<f>x)] . a = (ix) (<f>x) ; 

and p.D.a^ (ix) (<f>x) 

will mean p.D. [(ix) (<f>x)] . ~ [a = (ia>) (<£#)} ; 

but ^.D.^{a = (?*)(^r)} 

will mean p . D . ~ {[(?#) (</>#)] . a = (ix) (<f>x)}. 

This convention enables us, in the vast majority of cases that actually 
occur, to dispense with the explicit indication of the scope of a descriptive 
symbol; and it will be found that the convention agrees very closely with the 
tacit conventions of ordinary language on this subject. Thus for example, if 
"0)(W is " the so-and-so," " a ^ (ix) (fa)" is to be read "a is not the 
so-and-so," which would ordinarily be regarded as implying that " the so-and- 
so" exists; but "~ {a = (ix) (<}>x)} " is to be read "it is not true that a is the 
so-and-so," which would generally be allowed to hold if " the so-and-so " Hoes 
not exist. Ordinary language is, of course, rather loose and fluctuating in its 
implications on this matter; but subject to the requirement of definiteness, 
our convention seems to keep as near to ordinary language as possible. 

In the case when the smallest proposition enclosed in dots or other 
brackets contains two or more descriptions, we shall assume, in the absence 
of any indication to the contrary, that one which typographically occurs 
earlier has a larger scope than one which typographically occurs later. Thus 

(ix) (<\>x) = (ix) (yfrx) 
will mean (gc) : (f>x . = x . x = c i [(ix) (yfrx)'] . c = (ix) (tyx), 

while (ix) (yjrx) =* (ix) (<f>x) 

will mean (gd) : yjrx . = x . x = d : [(ix) (<f>x)~\ . (ix) (<f>x) = d. 

These two propositions are easily shown to be equivalent. 

(2) Classes. The symbols for classes, like those for descriptions, are, in 
our system, incomplete symbols : their uses are defined, but they themselves 
are not assumed to mean anything at all. That is to say, the uses of such 



72 INTRODUCTION [CHAP. 

symbols are so defined that, when the definienste substituted for the definimdum, 
there no longer remains any symbol which could be supposed to represent 
a class. Thus classes, so far as we introduce them, are merely symbolic or 
linguistic conveniences, not genuine objects as their members are if they are 
individuals. 

It is an old dispute whether formal logic should concern itself mainly with 
intensions or with extensions. In general, logicians whose training was mainly 
philosophical have decided for intensions, while those whose training was 
mainly mathematical have decided for extensions. The facts seem to be that, 
while mathematical logic requires extensions, philosophical logic refuses to 
supply anything except intensions. Our theory of classes recognizes and 
reconciles these two apparently opposite facts, by showing that an extension 
(which is the same as a class) is an incomplete symbol, whose use always 
acquires its meaning through a reference to intension. 

In the case of descriptions, it was possible to prove that they are in- 
complete symbols. In the case of classes, we" do not know of any equally 
definite proof, though arguments of more or less cogency can be elicited from 
the ancient problem of the One and the Many* It is not necessary for our 
purposes, however, to assert dogmatically that there are no such things as 
classes. It is only necessary for us to show that the incomplete symbols 
which we introduce as representatives of classes yield all the propositions for 
the sake of which classes might be thought essential. When this has been 
shown, the mere principle of economy of primitive ideas leads to the non- 
introduction of classes except as incomplete symbols,- 

,To explain the theory of classes, it is necessary first to explain the dis- 
tinction between extensional and intensional functions. This is effected by 
the following definitions : 

The truth-value of a proposition is truth if it is true, and falsehood if it is 
false. (This expression is due to Frege.) 

Two propositions are said to be equivalent when they have the same truth- 
value, i.e. when they are both true or both false. 

Two propositional functions, are said to he formally equivalent when they 
are equivalent with every possible argument, i.e. when any argument which 
satisfies the one satisfies the other, and vice versa. Thus "ob is a man " is 
formally equivalent to "x is a featherless biped"; "£ is an even prime" is 
formally equivalent to "ob is identical with 2." 

A function of a function is called extensional when its truth-value with any 
argument is the same as with any formally equivalent argument. That is to 

* Briefly, these arguments reduce to the following : If there is such an object as a class, it 
must be in some sense one object. Yet it is only of classes that many can be predicated. Hence, 
if we admit classes as objects, we must suppose that the same object can be both one and many, 
which seems impossible. 



Ill] EXTENSIONAL FUNCTIONS OF FUNCTIONS 73 

say, f(<f>z) is an extensional function of <f>z if, provided yfrz is formally equiva- 
lent to <f&, f(<f>z) is equivalent to f($$). Here the apparent variables <f> and 
ijr are necessarily of the type from which arguments can significantly be 
supplied to/. We find no need to use as apparent variables any functions 
of non-predicative types; accordingly in the sequel all extensional functions 
considered are in fact functions of predicative functions*. 

A function of a function is called intensional when it is not extensional. 

( The nature and importance of the distinction between intensional and 
extensional functions will be made clearer by some illustrations. The pro- 
position "'as is a man' always implies 'or is a mortal'" is an extensional function 
of the function "& is a man," because we may substitute, for "x is a man," 
"# is a featherless biped," or any other statement which applies to the same 
objects to which "a? is a man " applies, and to no others. But the proposition 
"A believes that 'x is a man' always implies l x is a mortal'" is an intensional 
function of "& is a man," because A may never have considered the question 
whether featherless bipeds are mortal, or may believe wrongly that there are 
featherless bipeds which are not mortal. Thus even if "x is a featherless 
biped" is formally equivalent to "x is a man," it by no means follows that a 
person who believes that all men are mortal must believe that all featherless 
bipeds are mortal, since he may have never thought about featherless bipeds, 
or have supposed that featherless bipeds were not always men. Again the 
proposition " the number of arguments that satisfy the function <f> ! z is n " is 
an extensional function of <f> ! z, because its truth or falsehood is unchanged if 
we substitute for <£!§ any other function which is true whenever tj>lz is true, 
and false whenever <f>\z is false. But the proposition "A asserts that the 
number of arguments satisfying <f> I z is n" is an intensional function of $ ! 2, 
since, if A asserts this concerning <p I $, he certainly cannot assert it concerning 
all predicative functions that are equivalent to <f> ! z, because life is too short. 
Again, consider the proposition " two white men claim to have reached the 
North Pole." This proposition states "two arguments satisfy the function 
'a? is a white man who claims to have reached the North Pole.'" The truth or 
falsehood of this proposition is unaffected if we substitute for "& is a white 
man who claims to have reached the North Pole " any other statement which 
holds of the same arguments, and of no others. Hence it is an extensional 
function. But the proposition "it is a strange coincidence that two white 
men should claim to have reached the North Pole," which states "it is a 
strange coincidence that two arguments should satisfy the function '& is a 
white man who claims to have reached the North Pole,'" is not equivalent to 
"it is a strange coincidence that two arguments should satisfy the function 
'ab is Dr Cook or Commander Peary.'" Thus "it is a strange coincidence that 
<f*t tb should be satisfied by two arguments" is an intensional function of <f>lx. 

* Cf. p. 53. 



.74 INTRODUCTION [CHAP. 

The above instances illustrate the fact that the functions of functions with 
which mathematics is specially concerned are extensional, and that intensional 
functions of functions only occur where non-mathematical ideas are introduced, 
such as what somebody believes or affirms, or the emotions aroused by some 
fact. Hence it is natural, in a mathematical logic, to lay special stress on 
extensional functions of functions. 

When two functions are formally equivalent, we may say that they have 
the same extension. In this definition, we are in close agreement with usage. 
We do not assume that there is such a thing as an extension: we merely 
define the whole phrase " having the same extension." We may now say that 
an extensional function of a function is one whose truth or falsehood depends 
only upon the extension of its argument. In such a case, it is convenient to 
regard the statement concerned as being about the extension. Since exten- 
sional functions are many and important, it is natural to regard the extension 
as an object, called a class, which is supposed to be the subject of all the 
equivalent statements about various formally equivalent functions. Thus 
e.g. if we say " there were twelve Apostles," it is natural to regard this state- 
ment as attributing the property of being twelve to a certain collection of 
men, namely those who were Apostles, rather than as attributing the property 
of being satisfied by twelve arguments to the function "& was an Apostle." 
This view is encouraged by the feeling that there is something which is 
identical in the case of two functions which " have the same extension." And 
if we take such simple problems as " how many combinations can be made of 
n things ? " it seems at first sight necessary that each " combination " should 
be a single object which can be counted as one. This, however, is certainly 
not necessary technically, and we see no reason to suppose that it is true 
philosophically. The technical procedure by which the apparent difficulty is 
overcome is as follows. 

We have seen that an extensional function of a function may be regarded 
as a function of the class determined by the argument-function, but that an 
intensional function cannot be so regarded. In order to obviate the necessity 
of giving different treatment to intensional and extensional functions of 
functions, we construct an extensional function derived from any function of 
a predicative function yfr ! z , and having the property of being equivalent to 
the function from which it is derived, provided this function is extensional, 
as well as the property of being significant (by the help of the systematic 
ambiguity of equivalence) with any argument d>z whose arguments are of the 
same type as those of yjr I z. The derived function, written "f {z(<f>z)} ," is de- 
fined as follows: Given a function /(^ ! z), our derived function is to be "there 
is a predicative function which is formally equivalent to <f>z and satisfies/." 
If <f)Z is a predicative function, our derived function will be true whenever 
f{4>z) is true. If f(<j>z) is an extensional function, and <j>z is a predicative 



Ill] DEFINITION OF CLASSES 75 

function, our derived function will not be true unless f(<f>z) is true; thus in 
this case, our derived function is equivalent to /(<££). If f(<f>2) is not an ex- 
tensional function, and if <f& is a predicative function, our derived function 
may sometimes be true when the original function is false. But in any case the 
derived function is always extensional. 

In order that the derived function should be significant for any function 
<f>z, of whatever order, provided it takes arguments of the right type, it is 
necessary and sufficient that/(^ \T) should be significant, where i/r ! £ is any 
predicative function. The reason of this is that we only require, concerning 
an argument <f>z, the hypothesis that it is formally equivalent to some predi- 
cative function yfrlz, and formal equivalence has/^he same kind of systematic 
ambiguity as to type that belongs to truth and falsehood, and can therefore 
hold between functions of any two different orders, provided the functions 
take arguments of the same type. Thus by means of our derived function we 
have not merely provided extensional functions everywhere in place of in- 
tensional functions, but we have practically removed the necessity for con- 
sidering differences of type among functions whose arguments are of the same 
type. This effects the same kind of simplification in our hierarchy as would 
result from never considering any but predicative functions. 

Iff(yfr I z) can be built up by means of the primitive ideas of disjunction, 
negation, (#).<£#, and (g#) . <£#, as is the case with all the functions of 
functions that explicitly occur in the present work, it will be found that, in 
virtue of the systematic ambiguity of the above primitive ideas, any function 
<f>z whose arguments are of the same type as those of f !| can significantly 
be substituted for -yfr I z in / without any other symbolic change. Thus in 
such a case what is symbolically, though not really, the same function f can 
receive as arguments functions of various different types. If, with a given 
argument <f>z, the function f(<f>z), so interpreted, is equivalent to f(yfr ! z) 
whenever yjr I z~ is formally equivalent to <f>z, then/ (2 ($2)} is equivalent to 
f(<j>z) provided there is any predicative function formally equivalent to <f>z. 
At this point, we make use of the axiom of reducibility, according to which 
there always is a predicative function formally equivalent to <f>%. 

As was explained above, it is convenient to regard an extensional function 
of a function as having for its argument not the function, but the class de- 
termined by the function. Now we have seen that our derived function is 
always extensional. Hence if our original function was f(yfr I z), we write the 
derived functiorr/{^(^)}, where "z(<f>z)" may be read " the class of arguments 
which satisfy <f>z," or more simply "the class determined by <f>z." Thus 
"f{2 (4>z)}" will mean: " There is a predicative function yjr ! z which is formally 
equivalent to <j>% and is such that/(i/r \z) is true." This is in reality a function 
of <f>%, but we treat it symbolically as if it had an argument z (<f>z). By the 
help of the axiom of reducibility, we find that the usual properties of classes 



76 INTRODUCTION [CHAP. 

result. For example, two formally equivalent functions determine the same 
class, and conversely, two functions which determine the same class are formally 
equivalent. Also to say that x is a member of 2 (<f>z), i.e. of the class determined 
by $%, is true when $% is true, and false when <f>a; is false. Thus all the 
mathematical purposes for which classes might seem to be required are fulfilled 
by the purely symbolic objects 2(<fnz), provided we assume the axiom of 
reducibility. 

In virtue of the axiom of reducibility, if <j>2 is any function, there is 
a formally equivalent predicative function -tyllz', then the class z~(<f>z) is 
identical with the class 2 (^ ! 2), so that every class can be defined by a 
predicative function. Hence the totality of the classes to which a given term 
can be significantly said to belong or not to belong is a legitimate totality, 
although the totality of functions which a given term can be significantly 
said to satisfy or not to satisfy is not a legitimate totality. The classes to 
which a given term a belongs or does not belong are the classes defined by 
a-functions; they are also the classes defined by predicative a-functions. Let 
us call them a-classes. Then "a-classes " form a legitimate totality, derived 
from that of predicative a-functions. Hence many kinds of general state- 
ments become possible which would otherwise involve vicious-circle paradoxes. 
These general statements are none of them such as lead to contradictions, and 
many of them such as it is very hard to suppose illegitimate. The fact that 
they are rendered possible by the axiom of reducibility, and that they would 
otherwise be excluded by the vicious-circle principle, is to be regarded as an 
argument in iavour of the axiom of reducibility. 

The above definition of " the class defined by the function cf>z," or rather, 
of any proposition in which this phrase occurs, is, in symbols, as follows: 

/{S^}.-:^):^.^.^!*:/^*}. Df. 
In order to recommend this definition, we shall enumerate five requisites 
which a definition of classes must satisfy, and we shall then show that the 
above definition satisfies these five requisites. 

We require of classes, if they are to serve the purposes for which they are 
commonly employed, that they shall have certain properties, which may be 
enumerated as follows. (1) Every prepositional function must determine a 
class, which may be regarded as the collection of all the arguments satisfying 
the function in question. This principle must hold when the function is 
satisfied by an infinite number of arguments as well as when it is satisfied by 
a finite number. It must hold also when no arguments satisfy the function ; 
i.e. the "null-class " must be just as good a class as any other. (2) Two pro- 
positional functions which are formally equivalent, i.e. such that any argument 
which satisfies either satisfies the other, must determine the same class; that 
is to say, a class must be something wholly determined by its membership, so 
that e.g. the class "featherless bipeds " is identical with the class M men," and 



Ill] CLASSES 77 

the class " even primes " is identical with the class " numbers identical with 2." 
(3) Conversely, two propositional functions which determine the same class 
must be formally equivalent; in other words, when the class is given, the 
membership is determinate : two different sets of objects cannot yield the same 
class. (4) In the same sense in which there are classes (whatever this sense 
may be), or in some closely analogous sense, there must also be classes of 
classes. Thus for example " the combinations of n things m at a time," where 
the n things form a given class, is a class of classes; each combination of 
m things is a class, and each such class is a member of the specified set of 
combinations, which set is therefore a class whose members are classes. Again, 
the class of unit classes, or of couples, is absolutely indispensable; the former 
is the number 1, the latter the number 2. Thus without classes of classes, 
arithmetic becomes' impossible. (5) It must under all circumstances be 
meaningless to suppose a class identical with one of its own members. For if 
such a supposition had any meaning "a e a" would be a significant propositional 
function*, and so would "a~^ot." Hence, by (1) and (4), there would be a 
class of all classes satisfying the function "a ~ e a." If we call this class k, we 
shall have 

a e k . =a ■ a ~ e- a. 
Since, by our hypothesis, "k e k" is supposed significant, the above equivalence, 
which holds with all possible values of a, holds with the value k, i.e. 

K € K . ~ . K ~> 6 K. 

But this is a contradiction f. Hence "aea" and "a~ea" must always be 
meaningless. In general, there is nothing surprising about this conclusion, 
but it has two consequences which deserve special notice. In the first place, 
a class consisting of only one member must not be identical with that one 
member, i.e. we must not have i i x = x. For we have #ei'#,„an4 therefore, if 
x = i l x, we have i'x e i'x, which, we saw, must be meaningless. It follows that 
"x=v l x" must be absolutely meaningless, not simply false. In the second 
place, it might appear as if the class of all classes were a class, i.e. as if 
(writing "Cls" for " class") "Cls e Cls" were a true 'proposition. But this com- 
bination of symbols must be meaningless; unless, indeed, an ambiguity exists 
in the meaning of "Cls," so that, in "Cls e Cls," the first "Cls" can be supposed 
to have a different meaning from the second. 

As regards the above requisites, it is plain, to begin with, that, in accordance 
with our definition, every propositional function <j>z determines a class, ff^). 
Assuming the axiom of reducibility, there must always be true propositions 
about 1z(4>z), i.e. true propositions of the form / {2 (<£.?)}, For suppose (f>z is 
formally equivalent to tyl z, and suppose yfrlz satisfies some function /. Then 

* As explained in Chapter I (p. 25), "area" means "x is a member of the class a*" or, 
more shortly, "a; is an a." The definition of this expression in terms of our theosy of classes 
will be given shortly. 

f This is the second of the contradictions discussed at the end o£ Chapter II. 



78 INTRODUCTION [CHAP. 

2 (<f>z) also satisfies/. Hence, given any function $% there are true propositions 
of the form/ {2 (<£>£)}, i.e. true propositions in which "the class determined by 
<j>z" is grammatically the subject. This shows that our definition fulfils the 
first of our five requisites. 

The second and third requisites together demand that the classes z(<f>z) and 
z tyz) should be identical when, and only when, their defining functions are 
formally equivalent, i.e. that we should have 

Here the meaning of u ^(<f>z)-^(^z)" is to be derived, by means of a two- 
fold application of the definition of /{$ (<f>z)}, from the definition of 

" x lz = 6lX" 
whichis x !a = 6?!t. = :(/):/!%!^.D./!6'!^ Df 

by the general definition of identity. 

In interpreting. "^ (<f>z) = ^ (yjrz)," we will adopt the convention which we 
adopted in regard to (ix)(<j>x) and (ix)(^rx), namely that the incomplete symbol 
which occurs first is to have the larger scope. Thus £ (<f>z) — z (tyz) becomes, 
by our definition, 

(ax) : <l> x ■ =* • X ! x '• X 1 * =*%W Z )> 
which, by eliminating 2 (yjrz), becomes 

(ax) =■ </>«•=*• %'• ® ■- (a#) '• f x • =x ■ 01 x : x ! £= 0l£, 
which is equivalent to 

(3%. 0) : <f>x . = x . X l x : fx . = x . 6\ x : x ! z = 6\ z, 
which, again, is equivalent to 

(ax ) ' 4> x ■ = x • x ! x ' '• ^r x ■ - * ■ x ! x > 

which, in virtue of the axiom of reducibility, is equivalent to 

<f>x . = x . tyx. 
Thus our definition of the use of z (<j>z) is such as to satisfy the conditions (2) 
and (3) which we laid down for classes, i.e. we have 

h :. 2 ((f>z) = z (yjrz) . = : <£# . = x . fx. 
Before considering classes of classes, it will be well to define membership 
of a class, i.e. to define the symbol "xe%{<t>z)" which may be read "# is a 
member of the class determined by <f>z." Since this is a function of the form 
f\z (<f>z)}, it must be derived, by means of our general definition of such func- 
tions, from the corresponding function f{f\1t). We therefore put 

x e f ! z . = . ^ ! x Df. 
This definition is only needed in order to give a meaning to "xezitfrz)"; the 
meaning it gives is, in virtue of the definition of/ \z (<f>z)}, 

(>&<f):<t>y.=y.1rly:ylrlx. 
It thus appears that "xez(<j>z)" implies <f>x, since it implies fix, and fix 
is equivalent to <j>x; also, in virtue of the axiom of reducibility, <f>x implies 
"xez(<f>z) } " since there is a predicative function f formally equivalent to <f>, 



Ill] CLASSES 79 

and x must satisfy i^>, since x {ex hypothesi) satisfies <£. Thus in virtue of the 
axiom of reducibility we have 

I- m . x ez (<f>z) . = . <f)X, 
i.e. # is a member of the class z (<f>z) when, and only when, x satisfies the 
function <j> which defines the class. 

We have next to consider how to interpret a class of classes. As we have 
defined f{z(<f>z)}, we shall naturally regard a class of classes as consisting of 
those values of %(<f>z) which satisfy f{z(<f>z)}. Let us write a for z (<f>z)\ then 
we may write a (fa) for the class of values of a which satisfy fa*. We shall 
apply the same definition, and put 

F{&(fa)}. = -.(<&g):fp.^ fi .g\p-F{g\a) Df, 
where "/3" stands for any expression of the form z (yfrl z). 

Let us take "76 a (fa)" as an instance of F{a(fa)}. Then 

\-:.rye&(fa). = :(zg):f/3.= fi .gl/3:yefflZ. 

Just as we put xe-^Mz . = .ty\x Df, 

so we put 7 eg\ a . = . g\ 7 Df. 

Thus we find 

H :. 7 ea(/a) . = : (^) ://3 . =0 . #! £ :5 r! 7 . 

If we now extend the axiom of reducibility so as to apply to functions of 
functions, i.e. if we assume 

(30) :/Wr!S).s,.0l (*!*),■ 
we easily deduce 

»■ = (30) = /{*(*'■ *)} ■ ^ • 5". {*(*« *)}, 
i.e. H:to)://3.=0.<7!/3. 

Thus I- : 7 e a (fa) . = .fy. 

Thus every function which can take classes as arguments, i.e. every function 
of functions, determines a class of classes, whose members are those classes* 
which satisfy the determining function. Thus the theory of classes of classes 
offers no difficulty. 

We have next to consider our fifth requisite, namely that "z(<f>z)ez($z)" 
is to be meaningless. Applying our definition of/{£ ($z)\, we find that if this 
collection of symbols had a meaning, it would mean 

(a^) : $ x ' —* • ^ • x ' ^ • ^ e ty • ^> 
i.e. in virtue of the definition 

x e yjrl z . = . yfrl x Df, 
it would mean (g>/r) : <f>x . = x . yfr I x : yfr ! (yjr ! 2 ). g 

But here the symbol "tyl (i/r! z)" occurs, which assigns a function as argument 
to itself. Such a symbol is always meaningless, for the reasons explained at 
the beginning of Chapter II (pp. 38 — 41). Hence "z (<f>z) e z (<ftz)" is meaning- 
less, and our fifth and last requisite is fulfilled. 

* The use of a single letter, such as a or |3, to represent a variable class, will be further 
explained shortly. 



80 INTRODUCTION [CHAP. 

As in the case of f(ix)(<f>x), so in that of f{z(<f>z)}, there is an ambiguity 
as to the scope of 2 (<j>z) if it occurs in a proposition which itself is part Of a 
larger proposition. But in the case of classes, since we always have the axiom 
of reducibility, namely . , s . , , 

which takes the place of El(ix)(<f>x), it follows that the truth-value of any 
proposition in which "z{^>z) occurs is the same whatever scope we may give to 
z (<f>z), provided the proposition is an extensional function of whatever functions 
it may contain. Hence we may adopt the convention that the scope is to be 
always the smallest proposition enclosed in dots or brackets in which z (<f>z) 
occurs. If at any time a larger scope is required, we may indicate it by " [z(<f)z)] " 
followed by dots, in the same way as we did for [(?#)($#)]• 

Similarly when two class symbols occur, e.g. in a proposition of the form 
/ {z (cf)z), 1z (tyz)}, we need not remember rules for the scopes of the two symbols, 
since all choices give equivalent results, as it is easy to prove. For the pre- 
liminary propositions a rule is desirable, so we can decide that the class symbol 
which occurs first in the order of writing is to have the larger scope. 

The representation of a class by a single letter a can now be understood. 
For the denotation of a is ambiguous, in so far as it is undecided as to which 
of the symbols z ((f>z), z (^z), z (x z )> etc - i* i s to stand for, where $z, -fz, xz, 
etc. are the various determining functions of the class. According to the choice 
made, different propositions result. But all the resulting propositions are equi- 
valent by virtue of the easily proved proposition: 

"r :$*=.**. D ./{£ (<f>z)} =/{3 (**)}." 
Hence unless we wish to discuss the determining function itself, so that the 
notion of a class is really not properly present, the ambiguity in the denotation 
of a is entirely immaterial, though, as we shall see immediately, we are led to 
limit ourselves to predicative determining functions. Thus "/(a)," where a is a 
variable class, is really "f{$(4>z)\," where <j> is a variable function, that is, it is 

"(af 1 ) .(f>x= x flx.f{yjrl z}," 
where <f> is a variable function. But here a difficulty arises which is removed 
by a limitation to our practice and by the axiom of reducibility. For the deter- 
mining functions <f>z, tyz, etc. will be of different types, though the axiom of 
reducibility secures that some are predicative functions. Then, in interpreting 
a as a variable in terms of the variation of any determining function, we shall 
be led into errors unless we confine ourselves to predicative determining func- 
tions. These errors % especially arise in the transition to total variation (cf. 
pp. 15, 16). Accordingly 

/a-.(a^). *!*=.*!*. /{*!*} Df- 
It is the peculiarity of a definition of the use of a single letter [viz. a] for a 
variable incomplete symbol that it, though in a sense a real variable, occurs 
only in the definiendum, while "<}>," though a real variable, occurs only in the 
definiens. 



™] BELATIONS 81 

Thus «f&" stands for 

and "(a) ./a" stands for yiT ' 

if (*):(a*).*l*s.^!«./ftr!3}. w 
Accordingly, in mathematical reasoning, we can dismiss the whole apparatus 
of functions and think only of classes as "quasi-things," capable of immediate 
representation by a single name. The advantages are two-fold: (1) classes are 
determined by their membership, so that to one set of members there is one 
class, (2) the "type" of a class is entirely defined by the type of its members. 

Also a predicative function of a class can be defined thus 
fla=.(^).<f,lx= x ^lx.fl{^l^} Df. 
Thus a predicative function of a class is always a predicative function of any 
predicative determining function of the class, though the converse does not hold. 

(3) Relations. With regard to relations, we have a theory strictly analogous 
to that which we have just explained as regards classes. Relations in extension, 
like classes, are incomplete symbols, We require a division of functions of two 
variables into predicative and non-predicative functions, again for reasons which 
have been explained in Chapter II. We use the notation "<f>l(x,y)" for a 
predicative function of x and y. 

We use "<£!(£, y)" for the function as opposed to its values; and we use 
"xp<j>(x,y)" for the relation (in extension) determined by (f>(x,y). We put 

f{$y<f>(x,y)} . = : ( a t) x4>{x,y): = x>y . yfrl (x,y) :f^l(x,p)} Df. 
Thus even when/ty ! (x, §)} is not an extensional function of y\r,f{ot§<f> (x, y)} 
is an extensional function of <£. Hence, just as in the case of classes, we deduce 

^''%H{x,y) = x§^(x,y). = -.<f>{x,y).= x>y .^(x,y), 
i.e. a relation is determined by its extension, and vice versa. 
On the analogy of the definition of "x e t I z" we put 
x{+l($,y)}y. = .-fl(x,y) Df*. 

This definition, like that of "«ef! z" is not introduced for its own sake, 
but in order to give a meaning to 

oo{xg<j>{x,y)}y. 
This meaning, in virtue of our definitions, is 

Ot) : 4>(*>y) • =x,y .y\r\{x,y)ix {irl(x,y)} y, 
le - i^)-<i>{^y)-=x,y.^\(^y):^\{x,y), 

and this, in virtue of the axiom of reducibility 

"(at) = <f> (*, y) • =x, y ■ t J to y)>" 

is equivalent to <f>(x,y). 

Thus we have always 

I- :z {$§4>(x,y)} y . = . <f>(x,y). 

This definition raises certain questions as to the two senses of a relation, which are dealt 
with in *21. 

R&w I n 



82 INTRODUCTION [CHAP. 

Whenever the determining function of a relation is not relevant, we may 
replace xp4> (x, y) by a single capital letter. In virtue of the propositions given 

above, 

r- :. R — S . = : xRy .= x , y . xSy, 

h \.R = &§4>(x,y).= : xRy . = x>y .4>(x,y), 
and \-.R = $$(xRy), 

Classes of relations, and relations of relations, can be dealt with as classes 
of classes were dealt with above. 

Just as a class must not be capable of being or not being a member of itself, 
so a relation must neither be nor not be referent or relatum with respect to 
itself. This turns out to be equivalent to the assertion that 4> ! (x, fi) cannot 
significantly be either of the arguments x or y in <£ ! (as, y). This principle, again, 
results from the limitation to the possible arguments to a function explained 
at the beginning of Chapter II. 

We may sum up this whole discussion on incomplete symbols as follows. 
The use of the symbol "(ix)(4>x)" as if in "f(ix)(4>x)" it directly represented 
an argument to the function /2 is rendered possible by the theorems 
b:.El(ix)(4>x).D:(x).fx.D.f(lx)(4>x), 
b : (ix) (<f>x) = (ix) tyx) . D .f(ix) ((fas) =f(ix) (-fx), 
h : E ! (ix) (4>x) . D . (ix) (4>x) = (ix) (<j>x), 
b : (ix) (4>x) = (ix) (yfrx) . = . (ix) (ifrx) = (ix) (<f>x), 
b : (ix) (4>x) = (ix) (tyx) . (ix) (fx) = (ix) ( X x). D . (ix) (4>x) = (ix) (%«). 
The use of the symbol "x (4>x)" (or of a single letter, such as a, to represent 
such a symbol) as if, in "f{x (4>x)}," it directly represented an argument a to a 
function f% is rendered possible by the theorems 
b:(a).fa.D.f{x(4>x)}, 
b : x(4>x) = x(^x) . D .f{x(4>x)} =/{^(^x)} 
b . x (4>x) = x (4>x), 

b : x (<j>x) = £ (yfrx). = . & (tyx) = x (4>x), 
\-:x(4>x) = x(yfrx).x(yJrx) = x( X x).^.x(4>x)=:x(xx). 
Throughout these propositions the types must be supposed to be properly 
adjusted, where ambiguity is possible. 

The use of the symbol "£$ {<£ (x, y)} " (or of a single letter, such as R, to 
represent such a symbol) as if, in "f{xg<f>(x,y)}," it directly represented an 
argument R to a function fR, is rendered possible by the theorems 
\-:(R).fR.D.f{x§4>(x,y)}, 

b:tif/4> (x, y)^x§^ (x, y) . D ./{$$ 4> 0> y)) =ffi§ t 0> V))> 
\-.Zp4>(x,y) = x§4>(x,y), 

\-:x§4> (x, y) = xp yjr(x,y). = .x§ir (x, y) = xy 4>(x, y), 
b:$§<f> (x, y) = x$ ^ (x, y).xy^ (x, y) = %<y X 0> V) ■ 

0..x§4>(x,y) = xpx(®>y)' 



hi] incomplete symbols 83 

Throughout these propositions the types must be supposed to be properly 
adjusted where ambiguity is possible. 

It follows from these three groups of theorems that these incomplete 
symbols are obedient to the same formal rules of identity as symbols which 
directly represent objects, so long as we only consider the equivalence of the 
resulting variable (or constant) values of propositional functions and not their 
identity. This consideration of the identity of propositions never enters into 
our formal reasoning. 

Similarly the limitations to the use of these symbols can be summed up 
as follows. In the case of (ix) (<f>x), the chief way in which its incompleteness 
is relevant is that we do not always have 

(x).fx.3 .f(ix)(<f>x), 
i.e. a function which is always true may nevertheless not be true of (ix) (<f>x). 
This is possible because f(ix) (<j>x) is not a value of fx, so that even when all 
values of fx are true, f(ix) (<f)x) may not be true. This happens when (ix) ((f>x) 
does not exist. Thus for example we have (x) .x = x, but we do not have 

the round square = the round square. 
The inference (x) .fx . D .f{ix) (<f)x) 

is only valid when E ! (ix) (4>x). As soon as we know E I (ix) (<f>x), the fact that 
(ix)(<px) is an incomplete symbol becomes irrelevant so long as we confine 
ourselves to truth -functions* of whatever proposition is its scope. But even 
when E ! (ix) (<f>x), the incompleteness of (fx) (<f>x) may be relevant when we 
pass outside truth -functions. For example, George IV wished to know whether 
Scott was the author of Waverley, i.e. he wished to know whether a proposition 
of the form "c — (ix) (<f>x)" was true. But there was no proposition of the form 
"c = y" concerning which he wished to know if it was true. 

In regard to classes, the relevance of their incompleteness is somewhat 

different. It may be illustrated by the fact that we may have 

z(tf>z) = ty !-£ . z (<f>z) = x ! z 

without having yfr ! z — % ! z. 

For, by a direct application of the definitions, we find that 

h : z (<f>z) = yfr ! z . = - (f>x = x \fr ! x. 
Thus we shall have 

h :<f>x= x ^\x.<px~ x xl^.D.2 (<f>z) = yfrlz .z (<j>z) = % ! f , 
but we shall not necessarily have tylz^xlz under these circumstances, for 
two functions may well be formally equivalent without being identical; for 
example, 

x = Scott . '= x . x = the author of Waverley, 

but the function "2 = the author of Waverley" has the property that George IV 
wished to know whether its value with the argument "Scott" was true, whereas 

* Cf. p. 8. 

6—2 



84 INTRODUCTION [CHAP. Ill 

the function " 2 = Scott " has no such property, and therefore the two functions 
are not identical. Hence there is a propositional function, namely 

x = y.x = z."D.y = z, 
which holds without any exception, and yet does not hold when for x we 
substitute a class, and for y and z we substitute functions. This is only 
possible because a class is an incomplete symbol, and therefore "£(<£.?)= ^S 2" 
is not a value of " x = y" 

It will be observed that "Olz — ^lz" is not an extensional function of 
i/r ! z. Thus the scope of 2 (<f>z) is relevant in interpreting the product 

2(<f>z) = y}rlz.z(<f>z) = x l Z' 
If we take the whole of the product as the scope of z(<f>z), the product is 
equivalent to 

(a#) : <f>x = x e ! x . e i > = ^ i % . e \ z = x '■ % 

and this does imply yjr I z = % ! 2. 

We may say generally that the fact that z (<f>z) is an incomplete symbol 
is not relevant so long as we confine ourselves to extensional functions of 
functions, but is apt to become relevant for other functions of functions. 



PART I 

MATHEMATICAL LOGIC 



SUMMARY OF PART I 

In this Part, we shall deal with such topics as belong traditionally to 
symbolic logic, or deserve to belong to it in virtue of their generality. We 
shall, that is to say, establish such properties of propositions, propositional 
functions, classes and relations as are likely to be required in any mathematical 
reasoning, and not merely in this or that branch of mathematics. 

The subjects treated in Part I may be viewed in two aspects: (1) as a 
deductive chain depending on the primitive propositions, (2) as a formal calculus. 
Taking the first view first: We begin, in *1, with certain axioms as to deduction 
of one proposition or asserted propositional function from another. From these 
primitive propositions, in Section A, we deduce various propositions which are 
all concerned with four ways of obtaining new propositions from given proposi- 
tions, namely negation, disjunction, joint assertion and implication, of which 
the last two can be defined in terms of the first two. Throughout this first 
section, although, as will-be shown at the beginning of Section B, our proposi- 
tions, symbolically unchanged, will apply to any propositions as values of our 
variables, yet it will be supposed that our variable propositions are all what 
we shall call elementary propositions, i.e. such as contain no reference, explicit 
or implicit, to any totality. This restriction is imposed on account of the 
distinction between different types of propositions, explained in Chapter II of 
the Introduction. Its importance and purpose, however, are purely philosophical, 
and so long as only mathematical purposes are considered, it is unnecessary to 
remember this preliminary restriction to elementary propositions, which is 
symbolically removed at the beginning of the next section. 

Section B deals, to begin with, with the relations of propositions containing 
apparent variables {i.e. involving the notions of "all" or "some") to each other 
and to propositions not containing apparent variables. We show that, where 
propositions containing apparent variables are . concerned, we can define 
negation, disjunction, joint assertion and implication in such a way that their 
properties shall be exactly analogous to the properties of the corresponding 
ideas as applied to elementary propositions. We show also that formal im~ 
'plication, i.e. "(#). (fxcDyfrx" considered as a relation of $oc to -\Jr£, has many 
properties analogous to those of material implication, i.e. "p D q" considered as 
a relation of p and q. We then consider predicative functions and the axiom 
of reducibility, which are vital in the employment of functions as apparent 
variables. An example of such employment is afforded by identity, which 
is the next topic considered in Section B. Finally, this section deals with 
descriptions, i.e. phrases of the form "the so-and-so" (in the singular). It is 
shown that the appearance of a grammatical subject "the so-and-so "is deceptive, 



88 MATHEMATICAL LOGIC [PART 

and that such propositions, fully stated, contain no such subject, but contain 
instead an apparent variable. 

Section C deals with classes, and with relations in so far as they are analogous 
to classes. Classes and relations, like descriptions, are shown to be "incomplete 
symbols" (cf. Introduction, Chapter III), and it is shown that a proposition 
which is grammatically about a class is to be regarded as really concerned with 
a prepositional function and an apparent variable whose values are predicative 
propositional functions (with a similar result for relations). The remainder of 
Section C deals with the calculus of classes, and with the calculus of relations 
in so far as it is analogous to that of classes. 

Section D deals with those properties of relations which have no analogues 
for classes. In this section, a number of ideas and notations are introduced 
which are constantly needed throughout the rest of the work. Most of the 
properties of relations which have analogues in the theory of classes are compara- 
tively unimportant, while those that have no such analogues are of the very 
greatest utility. It is partly for this reason that emphasis on the calculus- 
aspect of symbolic logic has proved a hindrance, hitherto, to the proper develop- 
ment of the theory of relations. 

Section E, finally, extends the notions of the addition and multiplication of 
classes or relations to cases where the summands or factors are not individually 
given, but are given as the members of some class. The advantage obtained 
by this extension is that it enables us to deal with an infinite number of 
summands or factors. 

Considered as a formal calculus, mathematical logic has three analogous 
branches, namely (1) the calculus of propositions, (2) the calculus of classes, 
(3) the calculus of relations. Of these, (1) is dealt with in Section A, while 
(2) and (3), in so far as they are analogous, are dealt with in Section C. We 
have, for each of the three, the four analogous ideas of negation, addition, 
multiplication, and implication or inclusion. Of these, negation is analogous 
to the negative in ordinary algebra, and implication or inclusion is analogous 
to the relation " less than or equal to " in ordinary algebra. But the analogy 
must not be pressed, as it has important limitations. The sum of two pro- 
positions is their disjunction, the sum of two classes is the class of terms 
belonging to one or other, the sum of two relations is the relation consisting 
in the fact that one or other of the two relations holds. The sum of a class 
of classes is the class of all terms belonging to some one or other of the 
classes, and the sum of a class of relations is the relation consisting in the 
fact that some one relation of the class holds. The product of two pro- 
positions is their joint assertion, the product of two classes is their common 
part, the product of two relations is the relation consisting in the fact that 
both the relations hold. The product of a class of classes is the part common 
to all of them, and the product of a class of relations is the relation consisting 



I] THE LOGICAL CALCULUS 89 

in the fact that all relations of the class in question -hold. The inclusion of 
one class in another consists in the fact that all members of the one are 
members of the other, while the inclusion of one relation in another consists 
in the fact that every pair of terms which has the one relation also has the 
other relation. It is then shown that the properties of negation, addition, 
multiplication and inclusion are exactly analogous for classes and relations, 
and are, with certain exceptions, analogous to the properties of negation, ad- 
dition, multiplication and implication for propositions. (The exceptions arise 
chiefly from the fact that "p implies q" is itself a proposition, and can there- 
fore imply and be implied, while "a is contained in 0," where a and # are 
classes, is not a class, and can therefore neither contain nor be contained in 
another class 7.) But classes have certain properties not possessed by pro- 
positions: these arise from the fact that classes have not a two-fold division 
corresponding to the division of propositions into true and false, but a three- 
fold division, namely into (1) the universal class, which contains the whole of 
a certain type, (2) the null-class, which has no members, (3) all other classes, 
which neither contain nothing nor contain everything of the appropriate type. 
The resulting properties of classes, which are not analogous to properties of 
propositions, are dealt with in *24. And just as classes have properties not 
analogous to any properties of propositions, so relations have properties not 
analogous to any properties of classes, though all the properties of classes have 
analogues among relations. The special properties of relations are much more 
numerous and important than the properties belonging to classes but not to 
propositions. These special properties of relations therefore occupy a whole 
section, namely Section D. 



SECTION A 

THE THEORY OF DEDUCTION 

The purpose of the present section is to set forth the first stage of the 
deduction of pure mathematics from its logical foundations. This first stage 
is necessarily concerned with deduction itself, i.e. with the principles by which 
conclusions are inferred from premisses. If it is our purpose to make all our 
assumptions explicit, and to effect the deduction of all our other propositions 
from these assumptions, it is obvious that the first assumptions we need are 
those that are required to make deduction possible. Symbolic logic is often 
regarded as consisting of two coordinate parts, the theory of classes and the 
theory of propositions. But from our point of view these two parts are not 
coordinate ; for in the theory of classes we deduce one proposition from another 
by means of principles belonging to the theory of propositions, whereas in the 
theory of propositions we nowhere require the theory of classes. Hence, in a 
deductive system, the theory of propositions necessarily precedes the theory 
of classes. 

But the subject to be treated in what follows is not quite properly described 
as the theory of propositions. It is in fact the theory of how one proposition 
can be inferred from another. Now in order that one proposition may be 
inferred from another, it is necessary that the two should have that relation 
which makes the one a consequence of the other. When a proposition q is a 
consequence of a proposition p, we say that p implies q. Thus deduction 
depends upon the relation of implication, and every deductive system must 
contain among its premisses as many of the properties of implication as are 
necessary to legitimate the ordinary procedure of deduction. In the present 
section, certain propositions will be stated as premisses, and it will be shown 
that they are sufficient for all common forms of inference. It will not be shown 
that they are all necessary, and it is possible that the number of them might 
be diminished. All that is affirmed concerning the premisses is (1) that they 
are true, (2) that they are sufficient for the theory of deduction, (3) that we 
do not know how to diminish their number. But with regard to (2), there 
must always be some element of doubt, since it is hard to be sure that one 
never uses some principle unconsciously. The habit of being rigidly guided 
by formal symbolic rules is a safeguard against unconscious assumptions; but 
even this safeguard is not always adequate. 



*1. PRIMITIVE IDEAS AND PROPOSITIONS 

Since all definitions of terms are effected by means of other terms, every 
system of definitions which is not circular must start from a certain apparatus 
of undefined terms. It is to some extent optional what ideas we take as 
undefined in mathematics; the motives guiding our choice will be (1) to 
make the number of undefined ideas as small as possible, (2) as between two 
systems in which the number is equal, to choose the one which seems the 
simpler and easier. We know no way of proving that such and such a system 
of undefined ideas contains as few as will give such and such results*. Hence 
we can only say that, such and such ideas are undefined in such and such 
a system, not that they are indefinable. Following Peano, we shall call the 
undefined ideas and the undemonstrated propositions primitive ideas and 
primitive propositions respectively. The primitive ideas are explained by means 
of descriptions intended to point out to the reader what is meant; but the. 
explanations do not constitute definitions, because they really involve the ideas 
they explain. 

In the present number, we shall first enumerate the primitive ideas 
required in this section; then we shall define implication; and then we 
shall enunciate the primitive propositions required in this section. Every 
definition or proposition in the work has a number, for purposes of reference. 
Following Peano, we use numbers having a decimal as well as an integral 
part, in order to be able to insert new propositions between any two. A change 
in the integral part of the number will be used to correspond to a new 
chapter. Definitions will generally have numbers whose decimal part is less 
than '1, and will be usually put at the beginning of chapters. In references, 
the integral parts of the numbers of propositions will be distinguished by 
being preceded by a star; thus "fcl/Ol " will mean the definition or proposition 
so numbered, and " #1 " will mean the chapter in which propositions have 
numbers whose integral part is 1, i.e. the present chapter. Chapters will 
generally be called " numbers." 

Primitive Ideas. 

(1) Elementary propositions. By an "elementary " proposition we mean 
one which does not involve any variables, or, in other language, one which 
does not involve such words as " all," " some," " the " or equivalents for such 
words. A proposition such as " this is red," where " this " is something given 
in sensation, will be elementary. Any • combination of given elementary 
propositions by means of negation, disjunction or conjunction (see below) will 

* The recognized methods of proving independence are not applicable, without reserve, to 
fundamentals. Cf. Principles of Mathematics, % 17. What is there said concerning primitive 
propositions applies with even greater force to primitive ideas. 



92 



MATHEMATICAL LOGIC [PAST I 



be elementary. In the primitive propositions of the present number, and 
therefore in the deductions from these primitive propositions in #2 — *5, the 
letters p, q, r, s will be used to denote elementary propositions. 

(2) Elementary propositional functions. By an " elementary propositional 
function" we shall mean an expression containing an undetermined consti- 
tuent, i.e. a variable, or several such constituents, and such that, when the 
undetermined constituent or constituents are determined, i.e. when values are 
assigned to the variable or variables, the resulting value of the expression 
in question is an elementary proposition. Thus if p is an undetermined 
elementary proposition, " not-p " is an elementary propositional function. 

We shall show in *9 how to extend the results of this and the following 
numbers (#1 — #5) to propositions which are not elementary. 

(3) Assertion. Any proposition may be either asserted or merely con- 
sidered. If I say " Caesar died," I assert the proposition " Caesar died," 
if I say " « Caesar died ' is a proposition," I make a different assertion, and 
" Caesar died '' is no longer asserted, but merely considered. Similarly in a 
hypothetical proposition, e.g. " if a = b, then b = a" we have two unasserted 
propositions, namely "a = b" and "b=a," while what is asserted is that the 
first of these implies the second. In language, we indicate when a proposition 
is merely considered by " if so-and-so " or " that so-and-so " or merely by 
inverted commas. In symbols, if p is a proposition, p by itself will stand 
for the unasserted proposition, while the asserted proposition will be de- 
signated by 

"\-.p." 
The sign "h" is called the assertion-sign*; it may be read "it is true that" 
(although philosophically this is not exactly what it means). The dots after 
the assertion-sign indicate its range ; 'that is to say, everything following is 
asserted until we reach either an equal number of dots preceding a sign 
of implication or the end of the sentence. Thus " h : p . D . a " means " it is 
true that p implies q" whereas " I- .p . D h . q " means "p is true ; therefore 
q is truef." The first of these does not necessarily involve the truth either 
of p or of q, while the second involves the truth of both. 

(4) Assertion of a propositional function. Besides the assertion of 
definite propositions, we need what we shall call "assertion of a propositional 
function." The general notion of asserting any propositional function is 
not used until #9, but we use at once the notion of asserting various special 
elementary propositional functions. Let <£# be a propositional function whose 
argument is w; then we may assert </>«? without assigning a value to as. 
This is done, for example, when the law of identity is asserted in the form 
"A is A ." Here A is left undetermined, because, however A may be deter- 

* We have adopted both the idea and the symbol of assertion from Frege. 
t Cf . Principles of Mathematics, § 38. 



SECTION a] primitive ideas and propositions 93 

mined, the result will be true. Thus when we assert <f>x, leaving x undetermined, 
we are asserting an ambiguous value of our function. This is only legitimate 
if, however the ambiguity may be determined, the result will be true. Thus 
take, as an illustration, the primitive proposition #1*2 below, namely 

" V :pvp . D .p," 
i.e. " ( p or p' implies p." Here p may be any elementary proposition: by 
leaving p undetermined, we obtain an assertion which can be applied to any 
particular elementary proposition. Such assertions are like the particular 
enunciations in Euclid: when it is said "let ABG be an isosceles triangle; 
then the angles at the base will be equal," what is said applies to any isosceles 
triangle; it is stated concerning one triangle, but not concerning a definite 
one. All the assertions in the present work, with a very few exceptions, assert 
propositional functions, not definite propositions. 

As a matter of fact, no constant elementary proposition will occur in the 
present work, or can occur in any work which employs only logical ideas. 
The ideas and propositions of logic are all general : an assertion (for example) 
which is true of Socrates but not of Plato, will not belong to logic*, and if an 
assertion which is true of both is to occur in logic, it must not be made 
concerning either, but concerning a variable x. In order to obtain, in logic, 
a definite proposition instead of a propositional function, it is necessary to 
take some propositional function and assert that it is true always or some- 
times, i.e. with all possible values of the variable or with some possible value. 
Thus, giving the name "individual" to whatever there is that is neither 
a proposition nor a function, the proposition " every individual is identical 
with itself" or the proposition " there are individuals " will be a proposition 
belonging to logic. But these propositions are not elementary. 

(5) Negation. If p is any proposition, the proposition "not-p," or "p is 
false," will be represented by "~p." For the present,/) must be an elementary 
proposition. 

(6) Disjunction. If p and q are any propositions, the proposition " p orq," 
i.e. "either p is true or q is true," where the alternatives are to be not 
mutually exclusive, will be represented by 

" PVq " 
This is called the disjunction or the logical sum of p and q. Thus " ~pvq" 

will mean u p is false or q is true"; "~ (pvq)" will mean "it is false that 

either p or q is true," which is equivalent to "p and q are both false"; 

" ~ (~ p v ~ q)" will mean "it is false that either p is false or q is false," which 

is equivalent to "p and q are both true " ; and so on. For the present, p and 

q must be elementary propositions. 

* When we say that a proposition "belongs to logic," we mean that it can be expressed in 
terms of the primitive ideas of logic. We do not mean that logic applies to it, for that would of 
course be true of any proposition. 



94 MATHEMATICAL LOGIC [PART I 

The above are all the primitive ideas required in the theory of deduction. 
Other primitive ideas will be introduced in Section B. 

Definition of Implication. When a proposition q follows from a proposition 
p, sp that, if p is true, q must also be true, we say that p implies q. The idea 
of implication, in the form in which we require it, can be defined. The mean- 
ing to be given to implication in what follows may at first sight appear some- 
what artificial; but although there are other legitimate meanings, the one here 
adopted is very much more convenient for Our purposes than any of its rivals. 
The essential property that we require of implication is this : "What is 
implied by a true proposition is true." It is in virtue of this property that 
implication yields proofs. But this property by no means determines whether 
anything, and if so what, is implied by a false proposition. What it does 
determine is that, if p implies q, then it cannot be the case that p is true and 
q is false, i.e. it must be the case that either p is false or q is true. The most 
convenient interpretation of implication is to say, conversely, that if either p 
is false or q is true, then " p implies q " is to be true. Hence "p implies q " 
is to be defined to mean : " Either p is false or q is true." Hence we put : 
*1'01. pDq. = . ~ pvq Df. 

Here the letters " Df " stand for " definition." They and the sign of equality 
together are to be regarded as forming one symbol, standing for " is defined 
to mean*." Whatever comes to the left of the sign of equality is defined to 
mean the same as what comes to the right of it. Definition is not among the 
primitive ideas, because definitions are concerned solely with the symbolism, 
not with what is symbolised ; they are introduced for practical convenience, 
and are theoretically unnecessary. 

In virtue of the above definition, when "p^q" holds, then either p is false 
or q is true ; hence if p is true, q must be true. Thus the above definition 
preserves the essential characteristic of implication ; it gives, in fact, the most 
general meaning compatible with the preservation of this characteristic. 

Primitive Propositions. 
#11. Anything implied by a true elementary proposition is true. Ppf. 

The above principle will be extended in #9 to propositions which are not 
elementary. It is not the same as "if ' p is true, then if p implies q, q is true." 
This is a true proposition, but it holds equally when p is not true and when p 
does not imply q. It does not, like the principle we are concerned with, enable 
us to assert q simply, without any hypothesis. We cannot express the principle 
symbolically, partly because any symbolism in which p is variable only gives 
the hypothesis that p is true, not the fact that it is true]:. 

* The sign of equality not followed by the letters "Df " will have a different meaning, to be 
denned later. 

t The letters "Pp" stand for "primitive proposition," as with Peano. 

t For further remarks on this principle, cf. Principles of Mathematics, § 38. 



SECTION A] PRIMITIVE IDEAS AND PROPOSITIONS 95 

The above principle is used whenever we have to deduce a proposition 
from a proposition. But the immense majority of the assertions in the 
present work are assertions of propositional functions, i.e. they contain an 
undetermined variable. Since the assertion of a propositional function is a 
different primitive idea from the assertion of a proposition, we require a 
primitive proposition different from *11, though allied to it, to enable us to 
deduce, the assertion of a propositional function " yfrx " from the assertions of 
the two propositional functions " (f>x" and "<f>xDyfrx" This primitive pro- 
position is as follows : 

*1'11. When <f>x can be asserted, where x is a real variable, and <f>x D yfrx can 
be asserted, where a; is a real variable, then yfrx can be asserted, where x is a 
real variable. Pp. 

This principle is also to be assumed for functions of several variables. 

Part of the importance of the above primitive proposition is due to the 
fact that it expresses in the symbolism a result following from the theory of 
types, which requires symbolic recognition. Suppose we have the two assertions 
of propositional functions " r- . <f>x " and " h . <j>x D yfrx " ; then the " x " in <f>x is 
not absolutely anything, but anything for which as argument the function "<j>x" 
is significant ; similarly in " <f>x D yfrx " the x is anything for which " <f>x D yfrx " 
is significant. Apart from some axiom, we do not know that the x's for which 
" <px D yfrx" is significant are the same as those for which " <f>x " is significant. 
The primitive proposition #111, by securing that, as the result of the assertions 
of the propositional functions "<f>x" and "<f>xDyfrx," the propositional function 
"yfrx" can also be asserted, secures partial symbolic recognition, in the form most 
useful in actual deductions, of an important principle which follows from the 
theory of types, namely that, if there is any one argument a for which both 
" (fta " and " yfra " are significant, then the range of arguments for which "<f>x " 
is significant is the same as the range of arguments for which " yfrx " is sig- 
nificant. It is obvious that, if the propositional function " <f)X D yfrx " can be 
asserted, there must be arguments a for which " <f>a D yfra " is significant, and 
for which, therefore, "<j>a" and "yfra" must be significant. Hence, by our 
principle, the values of x for which " <f>x " is significant are the same as those 
for which " yfrx " is significant, i.e. the type of possible arguments for <f>x (cf. 
p. 15) is the same as that of possible arguments for yfrx. The primitive pro- 
position #1'11, since it states a practically important consequence of this fact, 
is called the "axiom of identification of type." 

Another consequence of the principle that, if there is an argument a for 
which both <f>a and yfra are significant, then (px is significant whenever yfrx is 
significant, and vice versa, will be given in the " axiom of identification of real 
variables," introduced in #l - 72. These two propositions, #111 and #172, give 
what is symbolically essential to the conduct of demonstrations in accordance 
with the theory of types. 



96 MATHEMATICAL LOGIC [PART I 

The above proposition #1*11 is used in every inference from one asserted 
propositional function to another. We will illustrate the use of this proposition 
by setting forth at length the way in which it is first used, in the proof of 
#2 06. That proposition is 

"hz.pDq.D-.qDr .D.pDr." 
We have already proved, in #2 - 05, the proposition 

H :. q D r . D : p D q . D .p D r. 
It is obvious that #2'06 results from #205 by means of *204, which is 

\- :.p.D .qDr:D:q.D .pDr. 
For if, in this proposition, we replace p by q D>, q by p D q, and r by p D r, 
we obtain, as an instance of #204, the proposition 

\- "qO r .D :pD q .D .pDr :.D :.pD q .5 : q3 r .3 .pD r (1), 

and here the hypothesis is asserted by #2-05. Thus our primitive proposition 
#111 enables us to assert the conclusion. 

*1'2. h-.pvp.D.p Pp. 

This proposition states : " If either p is true or p is true, then p is true." 
It is called the "principle of tautology," and will be quoted by the abbreviated 
title of " Taut." It is convenient, for purposes .of reference, to give names to 
a few of the more important propositions; in general, propositions will be 
referred to by their numbers. 

#13. h : q . D . p v q Pp. 

This principle -states : "If q is true, then 'p or q is true." Thus e.g. if q is 
"•to-day is Wednesday" and p is "to-day is Tuesday," the principle states: 
" If to-day is Wednesday, then to-day is either Tuesday or Wednesday." It 
is called the " principle of addition," because it states that if a proposition is 
true, any alternative may be added without making it false. The principle 
will be referred to as "Add." 

*14. h:pvq.D.qvp Pp. 

This principle states that "p or q" implies "q or p." It states the 
permutative law for logical addition of propositions, and will be called the 
" principle of permutation." It will be referred to as " Perm." 

#15. h :pv(qvr). D .qv(pvr) Pp. 

This principle states : " If either p is true, or ' q or r ' is true, then either 
q is true, or 'p or r' is true." It is a form of the associative law for logical 
addition, and will be called the " associative principle." It will be referred to 
as "Assoc." The proposition 

pv(qvr) ,D .(pv q)vr, 
which would-be the natural form for the associative law, has less deductive 
power, and is therefore not taken as a primitive proposition. 



SECTION A] PRIMITIVE IDEAS AND PROPOSITIONS 97 

*1'6. \-: K q'2r.'5:pvq.'2.pvr Pp. 

This principle states : " If q implies r, then 'p or q ' implies ' p or r' " In 
other words, in an implication, an alternative may be added to both premiss 
and conclusion without impairing the truth of the implication. The principle 
will be called the "principle of summation," and will be referred to as "Sum." 
*1'7. If p is an elementary proposition, ~p is an elementary proposition. Pp. 

#1*71. If p and q are elementary propositions, p v q is an elementary pro- 
position. Pp. 

*1*72. If <f>p and yjrp are elementary propositional functions which take 
elementary propositions as arguments, <f>pvyfrp is an elementary propositional 
function. Pp. 

This axiom is to apply also to functions of two or more variables. It is 
called the " axiom of identification of real variables." It will be observed that 
if <£ and ^ are functions which take arguments of different types, there is no 
such function as " <f>a: v -tyx" because <j> and yjr cannot significantly have the 
same argument. A more general form of the above axiom will be given in #9. 

The use of the above axioms *l , 7 , 7l , 72 will generally be tacit. It is only 
through them and the axioms of #9 that the theory of types explained in the 
Introduction becomes relevant, and any view of logic which justifies these 
axioms justifies such subsequent reasoning as employs the theory of types. 

This completes the list of primitive propositions required for the theory 
of deduction as applied to elementary propositions. 



R&W I 



*2. IMMEDIATE CONSEQUENCES OF THE 
PRIMITIVE PROPOSITIONS 

Summary o/#2. 

The proofs of the earlier of the propositions of this number consist simply 
in noticing that they are instances of the general rules given in #1. In such 
cases,.these rules are not premisses, since they assert any instance of them- 
selves, not something other than their instances. Hence when a general rule 
is adduced in early proofs, it will be adduced in brackets*, with indications, 
when required, as to the changes of letters from those given in the rule to 

IV]) 

those in the case considered. Thus " Taut — — " will mean what " Taut " becomes 

P 

when <>jp is written in place of p. If " Taut — - " is enclosed in square brackets 

before an asserted proposition, that means that, in accordance with " Taut," 
we are asserting what "Taut" becomes when ~p is written in place of p. 
The recognition that a certain proposition is an instance of some general 
proposition previously proved or assumed is essential to the process of de- 
duction from general rules, but cannot itself be erected into a general rule, 
since the application required is particular, and no general rule can explicitly 
include a particular application. 

Again, when two different sets of symbols express the same proposition in 
virtue of a definition, say #1*01, and one of these, which we will call (1), has 
been asserted, the assertion of the other is made by writing "[(1).(*1*01)]" 
before it, meaning that, in virtue of *101, the new set of symbols asserts the 
same proposition as was asserted in (1). A reference to a definition is dis- 
tinguished from a reference to a previous proposition by being enclosed in 
round brackets. 

The propositions in this number are all, or nearly all, actually needed in 
deducing mathematics from our primitive propositions.. Although certain 
abbreviating processes will be gradually introduced, proofs will be given very 
fully, because the importance of the present subject lies, not in the propo- 
sitions themselves, but (1) in the fact that they follow from the primitive 
propositions, (2) in the fact that the subject is the easiest, simplest, and most 
elementary example of the symbolic method of dealing with the principles of 
mathematics generally. Later portions — the theories of classes, relations, 
cardinal numbers, series, ordinal numbers, geometry, etc. — all employ the 
same method, but with an increasing complexity in the entities and functions 
considered. 

* Later on we shall cease to mark the distinction between a premiss and a rule according to 
which an inference is conducted. It is only in early proofs that this distinction is important. 



SECTION A] IMMEDIATE CONSEQUENCES 99 

The most important propositions proved in the present number are the 
following : 

*202. h-.q.l.plq 

I.e. q implies that p implies q, i.e. a true proposition is implied by any 
proposition. This proposition is called the " principle of simplification " (re- 
ferred to as " Simp "), because, as will appear later, it enables us to pass from 
the joint assertion of q and p to the assertion of q simply. When the special 
meaning which we have given to implication is remembered, it will be seen 
that this proposition is obvious. 

*203. b'.pZ^q.Z.qD^p 

*215. h: ~p"5q m 3. ~qOp 

*216. h zpDq.D. ~gO ~p 

*217. h: ~gO -^p.D.p'Dq. 

These four analogous propositions constitute the "principle of transposition," 
referred to as " Transp." They lead to the rule that in an implication the two 
sides may be interchanged by turning negative into positive and positive into 
negative. They are thus analogous to the algebraical rule that the two sides 
of an equation may be interchanged by changing the signs. 

*204. \-:.p.0.q^r:D:q.0.pDr 

This is called the " commutative principle " and referred to as " Comm." 
It states that, if r follows from q provided p is true, then r follows from p 
provided q is true. 

*205. H:.gOr.D:jpDg.D.jOr 
*2;06. \-:.p3q.D:qDr.3.p3r 

These -two propositions are the source of the syllogism in Barbara (as will 
be shown later) and are therefore called the "principle of the syllogism" 
(referred to as " Syll "). The first states that, if r follows from q, then if q 
follows from p, r follows from p. The second states the same thing with the 
premisses interchanged. 

*208. b.pDp 

I.e. any proposition implies itself. This is called the " principle of identity " 
and referred to as " Id." It is not the same as the " law of identity " (" x is 
identical with x"), but the law of identity is inferred from it (cf. #1315). 

*2'21. h: ~p.O .pDq 

I.e. a false proposition implies any proposition. 

The later propositions of the present number are mostly subsumed under 
propositions in #3 or #4, which give the same results in more compendious 
forms. We now proceed to formal deductions. 

7—2 



100 



MATHEMATICAL LOGIC 



[PART I 



h :. q D r . D : ~p v q . D . ~p v r 



(1) 



(1) 



*201. \-:pD ~p.D . ~p 

This proposition states that, if p implies its own falsehood, then p is false* It 
is called the "principle of the reductio ad absurdum," and will be referred to as 
' Abs."* The proof is as follows (where "Bern" is short for " demonstration"): 

Dem. 

Taut ^±- V : ~£> v ~jp . D . ~p 

[(l).(*r01)] . h : p D ~p . D . ~.p 

*202. i-:g.3.;Og 

Dem. 

Add ^- H : q . D . ~ p v a 
L i> J 

[(1).(*1-01)] V-.q.D.pDq 

#203. h :/0~</. D.qO~p 

Dem. 

Perm — — — - H : ~r> v ~g . D . ~ov ~» 
L 2>» 2 J 

[(l).(*r01)] \-:pD~q.D.qD~p 

#204. h:.p.D.^Dr:D:^.D.pDr 

Dem. 

Assoc 21Pl2^3 h:.~i)v(~ovr).D.~ov(~»vr) 
P. ? J 
[(1).(*1-01)] h:.p.D.^Dr:D:g.D.^Dr 



(1) 



(1) 



(1) 



#205. b 
Dem. 

[Sum-P 

L p J 

[(l).(*r01)] h :. ? D r . D :pD # . 3 .p D r 
#206. hz.^O^.DigOr.D.pDr 
Dem. 

Comm<L :>r ' pDq ' p:>r ~\ br.qDr.D-.pDq.D.pDr:. 

p, q, r J 

D :.pDq'.D:qDr .D .pDr (1) 
[#2*05] F:.gDr.D:pD5.D.|)Dr (2) 

[(1).(2).*111] h:.pD^.D:^Dr.D.joDr 

In the last line of this proof, "(1) . (2) . #111" means that we are 
inferring in accordance with *ril, having before us a proposition, namely 
pDq.D-.qOr . Z> .pDr, which, by (1), is implied by q Dr. D :pDq . D .p Dr, 
which,, by (2), is true. In general, in such cases, we shall omit the reference 

to *rn. 

* There is an interesting historical article on this principle by Vailati, "A proposito d' un 
passo del Teeteto e di una dimostrazione di Euclide," Rivista di Filosofia e scienze affine, 1904. 



SECTION A] 



IMMEDIATE CONSEQUENCES 



101 



The above two propositions will both be referred to as the " principle of 
the syllogism " (shortened to " Syll "), because, as will appear later, the syllo- 
gism in Barbara is derived from them. 



*207. bzp.O.pvp *l-3 2 



Here we put nothing beyond "*1%3-," because the proposition to be 

proved is what #1*3 becomes when p is written in place of q. 

*208. h.jOp 

Dem. 

py 



*205 



-^M H ::p vp- 3 -p : 3 :. p ■ O.pvp: D •p'Sp 

h :pvp. D .p 

h :.p. "S.pvp : D -p^p 

\- m .p. D .pvp 

V .p^p 



[Taut] 

[(1).(2).*1-11] 

[*2-07] 

[(3).(4).*1-11] 
*21. b.~pvp [*2-08 . (*1-01)] 
*211. b.pv^p 

Dem. 



(1) 

(2) 
(3) 
(4) 



Perm 



>p,p 



[(1).*2-1.*1-11] \-.pv~p 
This is the law of excluded middle. 
*212. h.jO<^(~p) 
Dem. 



h : ~p vp . D ,p v*>p 



(1) 



r*2-n^ 

L p . 



(i) 



[(1).(*101)] h.pD~(~p) 

*213. Kjov~{~(~p)} 

This proposition is a lemma for *2'14, which, with #2*12, constitutes the 
principle of double negation. 

Dem. 



Sum 



'(~P)} 



[*2-12^] 
[(1).(2).*1'11] 

[(3).*2-n.*rii] 



1 

■ r- :. ~p . D . ~ (~(~p)} . D : 

I" : ~jp . D . ~{no(f^p)} 

h : p vr^p . D . p v~ (~(~jp)} 



(1) 

(2) 

(3) 



102 



MATHEMATICAL LOGIC 



[PABTI 



*214. K~(~p)3j? 

Bern. 

L q 



[(i).*2-i3.*rii] 

[(2).(*1-01)] \-.~(~p)Dp 

*215. b :~pDq .2 .cvq'Sp 
Dem. 



h :p v~{»j(~p)\ . D. ~{^(<^)} yp (l) 

K~{~(~/>)} Vp (2) 



h:.gD~(~g).D: ^jO^. D . ~pD~(~q) (1) 



^2-05^~ ( ~ g) 1 
_ P> r J 

*212 5] - 

Xl).(2).*lll] 

L 2 -03-^^1 
L P> 9. J 

j" cK 2*05 ~g»~<~P>>P ~| h .. ~(~p) Dp . D : ~ g D ~(~p) . D . ~g Dp 
r*2-05 ^^g'^P^^gX^g 3 "^^) "! h: . 



I- :™p"Dq . D . ™p 3~(~o/) 



(2) 
(3) 

(4) 

(5) 
(6) 



(7) 

(8) 
(9) 



p> g 

[(5).*2-14.*l-ll] 
*2'05 ~ P D q ' ~ P 3 ~(~g)> ~g 3~(~p) 

i>> g> »• 

~pD~(~g). D . ~gO~(~p) : D :. 

ojpDg'. D . ~/>D~(i~g') : D : r^pOq . D . <*-jqDe**>(~p) 
[(4).(7).*ril] h :. ~p Z> ? . D . ~p D ~(~?) : D : 

^jp D <7 . D . ~g D ~(~^>) 
[(3).(8).*111] I- : ~p D q . D . ~? D~(~p) 

#2*05 — *- — ^ — * - — — — - — £■ h ::~a D~(~») . D . ~qDp : 

I 2>, g> »" J 

D :.~pDq. D . <^gOoo(<->^p) : D :~pDq. D . ~gOp (10) 
[(6).(10).*1'11] h :. ~p D g . D . ~? 3 ~(~p) : D : 

~^0<7.D.~gOp (11) 
[(9).(11).*1-11] H ~_p D g . D . ~g Dp 

iV^ote on ^e proof of #2'15. In the above proof, it will be seen that (3), 
(4), (6) are respectively of the forms p^p 2 , p 2 Dp S) p^Pi, where j?iDp 4 is 
the proposition to be proved. From piOp 2 , p 2 "2p$, Ps^Pi the proposition 
p z yp 4 results by repeated applications of #2*05 or #2*06 (both of which are 
called " Syll "■). It is tedious and unnecessary to repeat this process every 
time it is used ; it will therefore be abbreviated into 

«[Syll]h.(a).(6).(o).Dh.(d), M 
where (a) is of the form p 1 Dp 2 , (b) of the form p 2 Op 3 , (c) of the form p 3 Dp 4 , 
and (d) of the form p x "Dp 4 . The same abbreviation will be applied to a sorites 
of any length. 



*2-03 — ^ 
9 



SECTION A] IMMEDIATE CONSEQUENCES 103 

Also where we have "b .p" and "b . pyDpz," and j» 2 is the proposition to 
be proved, it is convenient to write simply 

[etc.] K^," 

where " etc." will be a reference to the previous propositions in virtue of which 
the implication "pi^pa" holds. This form embodies the use of #1\L1 or #1*1, 
and makes many proofs at once shorter and easier to follow. It is used in the 
first two lines of the following proof 

*216. bip^q. D.~gO~p 

Dem. 

[*2\L2] f-.?D~(~?).D 

[*2-05] ' b:p5q.3.pD~(~q) (1) 

b :j?D~(~<7). D ,~qD~p (2) 

[Syll] b.(l).(2).Db:pDq.D.~qD~p 

Note. The proposition to be proved will be called "Prop," and when 
a proof ends, like that of *2'16, by an implication between asserted propo- 
sitions, of which the consequent is the proposition to be proved, we shall 
write " b . etc. D h . Prop". Thus 'OH. Prop " ends a proof, and more or less 
corresponds to "Q.E.D." 

*217. b i^q'Dr^p . D .p"5q 

Dem. 

r*2-03^^| b:~ q 3~p.3.p5~(~ q ) (l) 

[*214] b:~(~q)3q:3 

[*2-05] b:pD~(~q).O.pDq (2) 

[Syll] r.(l).(2).3h.Prop 

#2*15, *2'16 and #2*17 are forms of the principle of transposition, and will 
be all referred to as " Transp." 

*218. h:~pDp.D.p 

Dem. 

[*2-12] h.jO~(~p).D 

[*2-05] r-.~jpDp.D.~jO~(~p) (1) 

I- :~pD~(~p). D . ~(~|j) (2) 



[«*=*] 



[Syll] I- . (1) . (2) . D I- : ~p Dp . D . ~(~p) (3) 

[*2-14] b.~(~p)Dp (4) 

[Syll] K(3).(4).DKProp 

This is the complement of the principle of the reductio ad absurdum. It 



104 



MATHEMATICAL LOGIC 



[PART I 



states that a proposition which follows from the hypothesis of its own false- 
hood is true. 
*2'2. hzp.D.pvq 
Dem. 



r- . Add . D h : p ... D . q vp 
[Perm] r- : q vp . D . p v q 
[Syll] H . (1) . (2) . D h . Prop 



(1) 
(2) 



*221. h:~p.3.pl q |~*2-2^1 

The above two propositions are very frequently used. 
*2*24. h : p . D . ~p D q [*2*21 . Comm] 



*2'25. \~ z.pzv zpvq.D .q 

Dem. 

f- . *21 . D 1- : ~(p v q) . v . (p v q) : 

[Assoc] D 1- :^> . v . {~(p v q) . v . q) : D h . Prop 


*2-26. H:.~p:v:^D ? .D.g |*2-25 ^ 

*2-27. H.p.D:pOg.D.? [*2'26] 
*2'3. V : » v (o v r) . D . p v (r v o) 





Perm 



Dem. 

l^— H:<7vr.D.rv<7: 

Sum — — ' D I- : p v (q v r) . D . p v (r v q) 

*2'31. H : p v (g v r) . D . (p v q) v r 

This proposition and *2*32 together constitute the associative law for 
logical addition of propositions. In the proof, the following "abbreviation, 
(constantly used hereafter) will be employed*: When we have a series of 
propositions of the form aD6, 6 3c, cOd, all asserted, and "aDd" is the 
proposition to be proved, the proof in full is as follows : 

(- :. a D b . D : b D c . D . a D c 

h:a.D.& 

h : bDc. D .aDc 

H&.D.c 

bza.O.c 

:.aDc.D:cDrf.D.a!>^ 

:cDd. D . aDd 

:c.O .d 

:a .D .d 



[Syii] 

[(1).(2).*111] 

[(3).(4).*1'11] 

[Syll] 

[(5).(6).*111] 



[(7).(8).*111] 



(1) 
(2) 
(3) 
(4) 

(5) 
(6) 

(?) 
(8) 



* This abbreviation applies to the same type of cases as those concerned in the note to *2-15, 
but is often more convenient than the abbreviation explained in that note. 



IMMEDIATE CONSEQUENCES 



105 



SECTION A] 

It is tedious to write out this process in full.; we therefore write simply 

h:a.DJ. 

[etc.] D . c . 

[etc.] D.dzOh. Prop, 
where "aDd" is the proposition to be proved. We indicate on the left by 
references in square brackets the propositions in virtue of which the successive 
implications hold. We put one dot (not two) after " 6," to show that it is b, 
not " a D b," that implies c. But we put two dots after d, to show that now 
the whole proposition " a "2d" is concerned. If "a D d " is not the proposition 
to be proved, but is to be used subsequently in the proof, we put 

h za.D .b . 

[etc.] D .c . 

[etc.] D . d 

and then " (1) " means " a D d." The proof of *2*31 is as follows : 

Dem. 

[*2'3] h : p v (q v r) . D . p v (r v q) . 

Assoc — 



(1). 



i_ 9> r J 

fperm^^" 



D . r v (p v q) . 

D . ( p v q) v r : D I- . Prop 



I- : ( p v <?) v r . D . r v (p v 9) 

D .p v (g v r) : D I- . Prop 



L P> 2 

*2'32. I- :(pv^)vr.D.pv(q'vr) 

Dem. 

Perm- — — — 
# 9. 

Assoc ' ^' ^ 
p, q, r 

[*2-3] 

*2'33. pvqvr. = . (pvq)vr Df 

This definition serves only for the avoidance of brackets. 

#236. \- z.qDr .D zpw q .D .rvp 

Dem. 

[Perm] Vzpvr.D.rvpz 

\sy\\ pVq,pVr ' r * P ~\Dh:.pvq.D.pvr:D:pvq.3.rvp (1) 

[Sum] hz.qOr. "2 zpvq ."2 .pvr (2) 

h.(l).(2).Syll.Dh.Prop 
#237. h z.qOr .0 zqvp .0 .pvr 

[Syll . Perm . Sum] 
*2'38. hz.qOr.Dzqvp.O.rvp 

[Syll . Perm . Sum] 



106 MATHEMATICAL LOGIC [PART I 

The proofs of #2*37 -38 are exactly analogous to that of #2*36. (We use 
" *2'37'38 " as an abbreviation for " *2*37 and *2"38." Such abbreviations will 
be used throughout.) 

The use of a general principle of deduction, such as either form of " Syll," 
in a proof, is different from the use of the particular premisses to which the 
principle of deduction is applied. The principle of deduction gives the general 
rule according to which the inference . is made, but is not itself a premiss in 
the inference. If we treated it as a premiss, we should need either it or some 
other general rule to enable us to infer the desired conclusion, and thus we 
should gradually acquire an increasing accumulation of premisses without 
ever being able to make any inference. Thus when a general rule is adduced 
in drawing an inference, as when we write " [Syll] h . (1) . (2) . D I- . Prop," the 
mention of " Syll " is only required in order to remind the reader how the 
inference is drawn. 

The rule of inference may, however, also occur as one of the ordinary 
premisses, that is . to say, in the case of " Syll " for example, the proposition 
"p D q . D : q D r . D . p D r " may be one of those to which our rules of deduction 
are applied, and it is then an ordinary premiss. The distinction between the 
two uses of principles of deduction is of some philosophical importance, and 
in the above proofs we have indicated it by putting the rule of inference in 
square brackets. It is, however, practically inconvenient to continue to dis- 
tinguish in the manner of the reference. We shall therefore henceforth both 
adduce ordinary premisses in square brackets where convenient, and adduce 
rules of inference, along with other propositions, in asserted premisses, i.e. we 
shall write e.g. 

"h.(l).(2).Syll.Db.Prop" 
rather than " [Syll] h . (1) . (2) . D h . Prop " 

*2*4. H :.p. v .pyq ' D -pvq 

Dem. 

h .#231 . D h \.p. v\.p v q : D :pvp . v . q : 

[Taut.*2-38] D:pvq:.Dh. Prop 

*2'41. \-:.q.v.pvq:'D.pvq 

Dem. 

Assoc "tliA 
p, q, r 

[Taut.Sum] D : p v.q :. D h . Prop 

*2'42. f- :. ~p .w .pDq:D . pOg #2-4 — - 

*2-43. b:.p.D.pDq:0.pDq [*2"42] 
*2"45. h:~(pvq).D. ~p [*2'2 . Transp] 

*2'46. h : ~ (p v q) . D . ~ q [*1 -3 . Transp] 



b:.q.v.pvq:0:p.v.qvq: 



IMMEDIATE CONSEQUENCES 



*2-45.*2'2-^.Sylf| 



*2'46 . *l-3 
*2'45 . *22 

•2-47 ^£' 

PA 



<**>p, <~ 

p> <1 



■Syll] 



■Syll] 



*2-48 



P 



P 



SECTION A] 

#247. \-:~(pvq).0.<s>pvq 

*2*48. f-:~(pvg).D.pv~# 

*2*49. H:~(pv<7).0.~pv<>^ 

*2'5. I- :«^(pDgr). D.«yp>gr 

*2'51. r-:«%»(jOg).D.|0«^g' 

*2'52. >:~(^02).D.~pO~2 *2'49 -^ 

*2 521. h:~(jOg)'.D.gOj>' [*2'5217] 

*2'53. hzpv^.D.^^Og' 

h .*2-12-38 . D I- :pv^ . D . ~(~jp) vg : D h .Prop 
H^jO^.D.jovg [*214-38] 

h:.~j». Drjjvg.D .g [#2*53 . Comm] 

h:.~g . D zjjvg.D.jp #2*55 ^: . Perm 
h :. ~p D g . D : p D q . D . q 



107 



*254. 
*255. 

*256. 



*26. 



Bern. 



[*2'38] h:.~pDg'.D:~pvg'.D.g'V(/ 

[Taut . Syll] r :.~p v g . 3 . q v g : D :~p v # . D . <? 

H . (1) . (2) . Syll . D h :.~p D g . D :~p v g . D . q :. D H . Prop 



(1) 
(2) 



*2'61. h 

*262. h 

*2621. 1- 

*2-63. h 

*264. H 



.p"Dq.D: ~p D ^ . D .q 
• pvq- D:p^q.O .q 
.pOq.D :pvq. D .q 
. p v ^ . D : ~jp v ^ . D . ^ 

.^} v q. D :pv~^. D .^) 



*2*65. \-:.pDq.D:pD~q.D.r*>p 
*2 67. h:.|)Vf.D.jf':D.|)D9 



[*2-6 . Comm] 
[*2-53-6 . Syll] 
[*2'62 . Comm] 
[*2-62] 

T*2-63 &-? . Perm 
L P>2 

[*2-64^] 



Bern. 



[*2 - 54.Syll] h z.pvq.D ,q:D : ~j? D q . D . q 
[*2'24.Syll] \-:.~pDq.D.q:D.pDq 
1- . (1) . (2) . Syll . D h . Prop 



(1) 
(2) 



108 



MATHEMATICAL LOGIC 



[PART I 



*2'68. \r:.p2q.2.q:D.pvq 
Bern. 

1*2-67 ^2] hz.pOq.O.q-.^.^pOq 

h . (1) . *2*54 . D h . Prop 
*2-69. \-:.pOq.^.q:0'.qDp.^.p [*2-68 . Perm . *2'62 ^ 1 

*2'73. \-:.p3q.0:pvqvr.3.qvr [*2-621'38] 
*274. H.^Dp.D^vgvr.D.^vr *2'73 ?L& . Assoc . Syll 

*2*75. h::pvg.D:._p.v.?Dr:D.|)vr *2'74 ^ . *2'53'31 



(1) 



*2-74^2 
*276. h.p.v.prOspv^.D.pvr [*2'75 . Comm] 



*2-76 



2> J 



*277. hz.p.O.qlr-.lipDq.O.p'Dr ■ 
*2'8. r- :.gvr .D :~rvs. D.^vs 

h . *2 - 53 . Perm . D f- :. q v r . : ~r D g : 
[*2'38] D : ~r vs.D.gvsr.DH. Prop 

*2'81. I- :: q.D.rOszDz.pvq.Dzpvr.'S.pvs 

Bern. 
h.Sum . Obizq. D.rDs: D :.jpvg.D :p. v . Os 
> . *2-76 . Syll .Dhzzpvq.lzp.v.r'Ssz.Dz. 

K(l).(2).Dh.Prop 
*2'82. H.pvgvr .D:^v~rv$.D.jjvgvs 



(1) 
(2) 



*2'8 . *2-81 



o, r, s 



*2'83. H- :: jp . D . <j D r : D :. jp . D . r D * : D: ^> . D . gO s 

L 2 . 82 jm^l 
L p> ? J 

*285. l-:.f v^.3.|rvr : D :p. v.gOr 

[Add.Syll] K:.pvg.D.r:D.gDr (1) 

H . *2'55 . 3 H :: ~jp . D :.^> v r . D . r :. 
[Syll] Or.^vg. D.pvrz D :^vg .D. r:. 

[(1).*2'83] Oi.pyq.D.pvr'.OzqDr (2) 

H . (2) . Comm . D I- :. _pvg. D .pvr : D :~p. D .gOr : 
[#2'54] Oip.v.gOr:. Dh. Prop 

*2'86. :f:.j)Dg.>..|)Dr:3 : jp . 3 . 5 D r *2'85 -^ 1 



*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS 

Summary q/"#3. 

The logical product of two propositions p and q is practically the pro- 
position "p and q are both true." But this as it stands would have to be a 
new primitive idea. We therefore take as the logical product the proposition 
~(~^v~j), i.e. "it is false that either p is false or q is false," which is 
obviously true when and only when p and q are both true. Thus we put 

*301. p.q. = .~(~^v~g) Df 

where "p . q " is the logical product of p and q. 
*302. pDqDr. = .pDq.qDr Df 

This definition serves merely to abbreviate proofs. 

When we are given two asserted propositional functions " H . <f>x " and 
" H . •fyx" we shall have " L. <f>x . yjrx " whenever <f> and \fr take arguments of 
the same type. This will be proved for any functions in #9 ; for the present, 
we are confined to elementary propositional functions of elementary pro- 
positions. In this case, the result is proved as follows : 

By *1'7, ~(j>p and ~yjrp are elementary propositional functions, and there- 
fore, by #1*72, ~<|)pv~ yjrp is an elementary propositional function. Hence 

by #2-11, 

h:~^pv~i^p.v.< , v(~<^pv~^fp). 

Hence by #2-32 and *r01, 

h :. <f>p . D : yjrp . D . ~(~^,jo v o->yJfp), 
i.e. by *3'01, 

I- :. <f>p . D : i|rjp .D ,<j)p . typ. 

Hence by #1'11, when we have "h. <\>p" and "V .typ" we have "V .fyp.-typ." 
This proposition is #3*03. It is to be understood, like *1'72, as applying "also 
to functions of two or more variables. 

The above is the practically most useful form of the axiom of identification 
of real variables (cf. *1'72). In practice, when the restriction to elementary 
propositions and propositional functions has been removed, a convenient means 
by which two functions can often be recognized as taking arguments of the 
same type is the following : 

If <f)X contains, in any way, a constituent %(#, y, z, ...) and yjrx contains, 
in any way, a constituent %(#, u,v, ...), then both <f>x and yjrx take arguments 
of the type of the argument x in ^ (x, y,z, .. .), and therefore both <}>x and yjrx 
take arguments of the same type. Hence, in such a case, if both <j>x and yjrx 
can be asserted, so can <f>x . yfrx. 



110 MATHEMATICAL LOGIC [PART I 

As an example of the use of this proposition, take the proof of #3 "47. We 
there prove 

h z.p^r .qDs. D :p-q* D.g-.r (1) 

and b:.pDr.q'Ds. D:q.r. 3 .r '. s (2) 

and what we wish to prove is 

p'Dr.qOs.Ozp.q.O.r.s, 
which is #3*47. Now in (1) and (2), p, q, r, s are elementary propositions 
(as everywhere in Section A); hence by #l'7"7l, applied repeatedly, 
"p D r . q D s . D :p . q . D . q . r" and "p 3 r . q D s . D : q . r . D . r . s" are ele- 
mentary propositional functions. Hence by *3'03, we have 

\ m ::pDr.q'Ds.D:p.q.D.q.r:.pDr.qDs.D:q.r.D.r.s, 
whence the result follows by #3'43 and *3*33. 

The principal propositions of the present number are the following : 
*3'2. bz.p.D-.q.D.p.q 

I.e. "p implies that q implies p . q," i.e. if each of two propositions is true, 
so is their logical product. 

*3-26. V-.p.q.l.p 

*3'27. bip.q.D.q 

I.e. if the logical product of two propositions is true, then each of the two 
propositions severally is true. 

*3'3. \-:.p.q.D .r:D:p.D.q^r 

I.e. if p and q jointly imply r, then p implies that q implies r. This 
principle (following Peano) will be called "exportation," because q is "exported" 
from the hypothesis. It will be referred to as " Exp." 
*3'31. \-:.p.D.qDr:D:p.q.'D.r 

This is the correlative of the above, and will be called (following Peano) 
" importation " (referred to as " Imp "). 
*3-35. b-.p.p^q.D.q 

I.e. "ifp is true, and q follows from it, then q is true." This will be called 
the "principle of assertion" (referred to as "Ass"). It differs from #11 by 
the fact that it does not apply only when p really is true, but requires merely 
the hypothesis that p is true. 
*3'43. I- z.pDq.pDr . D :p . D .q.r 

I.e. if a proposition implies each of two propositions, then it implies their 
logical product. This is called by Peano the " principle of composition." It 
will be referred to as " Comp." 
*3*45. b z.pDq.^zp .r .D .q.r 

I.e. both sides of an implication may be multiplied by a common factor. 
This is called by Peano the " principle of the factor." It will be referred to 
as " Fact." 



SECTION A] THE LOGICAL PRODUCT OP TWO PROPOSITIONS 



111 



♦347. Hr.pDr.gOff.Drp.g.D.r.f 

I.e. if p implies q and r implies s, then p and q jointly imply r and s 
jointly. The law of contradiction, " h . ~ (p . ~p)," is proved in this number 
(♦3*24); but in spite of its fame we have found few occasions for its use. 

♦301. p,q. = .~(<^>pvn*>q) Df 

♦302. p3q0r. = .pDq.q0r Df 

♦3*03. Given two asserted elementary prepositional functions "\- .<f>p " and 

" h . yfrp" whose arguments are elementary propositions, Ave have h . <f>p . yfrp. 
Bern. 

V . *l'7-72 . *211 .DH:~$pv~<^- v -~(~# v ~ typ) (1) 

I- . (1) . #2*32 . (*1'01) . D h :. <j>p . D : yfrp . D . ~ (~ <f,p v ~ yjrp) (2) 
I- . (2) . (*3'01) .Dh:.^).D:^).D.#.^) (3) 

r . (3) . *1\L1 . D h . Prop 
hip.q.3.~(~pv~q) [Id . (♦3 , 01)] 



*31. 

♦3*11. h :~(~pv~^). D -p-q 



Id . (*3-01)] 
♦211 ~*> v ~g " 



♦311 . Transp] 



♦312. h : <>jp. v . "*>q . v .p . g 

♦3*13. H : <^ (^> . q) . D . ~p v ~ <? 

♦314. h : ~p v~q.D.~(p.q) [#3l . Transp] 

♦3 2. hz.p.l-.q.O.p.q [*3'12] 

♦3-21. h:.gr.D:^.D.p. ? [*3'2 . Comm] 

♦3-22. h'.p.q.D.q.p 

This is one form of the commutative law for logical multiplication. A 
more complete form is given in *4 3. 
Bern: 



V : ~ (q . p) . D . ~ q v ~p . 



Ls-13 ££ 

L i>> ?J 

[Perm] D.~pv~g. 

[#314] 2.~(p.q) 

h . (1) . Transp . D h . Prop 
Note that, in the above proof, " (1) " stands for the proposition 

w ~(3.p).3.-(p.j), M 
as was explained in the proof of *2-31. 
♦3'24. h.~(p.~p) 
Dem. 

r^p' 



(1) 



♦2-11 

P J 

♦3-14 --£ 



I - . < — ' ( JO . <^p) 



The above is the law of contradiction. 



112 



MATHEMATICAL LOGIC 



[PART I 



*3*26. bip.q.l.p 
Bern. 

*202 






[(1).(*1-01)J 
*2-31] 

*253 p 



q>p 



p> 

[(2).(*3-01)] 

*327. bzp.q.l.q 

Dem. 

[*3-22] 



H :p.D *q"2p 

p . v . ~qvp; 
pv~q . v .p: 

3 h ;<^>(~pv<^>q). D ,p 

h zp.q.O .p 



r: 



(1) 



(2) 



\- zp.q.'D.q.p. 

#3-26 2l21- D.grDKProp 

L . # SJ 
#3'26*27 will both be called the "principle of simplification," like *2*02, 

from which they are deduced. They will be referred to as " Simp." 
#33. h :.p .q ,D .r:D :p .D .qOr 

Dem. 

[Id.(*301)] \-:.p.q.O 
[Transp] 
[Id.(*l-01)] 
[Comm] 
[Transp.Syll] 
*331. hz.p.D.qDr-.'D-.p.q.D, 



D : ~ ( ™p v ~ q) . D . r : 

D : ~r . D .p D^g : 
D:j?.D.~rD~g': 
D : p . . q D r :. D h . Prop 



•Dem. 



Id.(*101)] h r.jp.D.^Dr :D:~^.v.~^vr : 
#2*31] D :~j3V~^. v.r : 

pv ~q 



*2-53 



2>> 






P : ~(~j9 v oj<2) . D . r : 
D:j).g.D.r:.DK Prop 



[Id.(*3'01)] 
*3 33. hzpDq.qDr.D .p Dr [Syll . Imp] 
*3-34. \-:qDr.pDq.D.pDr [Syll . Imp] 

These two propositions will .hereafter be referred to as "Syll"; they are 
usually more convenient than either #2'05 or #2*06. 
*335. hzp.pDq.l.q [*2"27 . Imp] 
#337. h :.p. q. D . r : D :p,<*>>r . D .~q 
Dem. 

h . Transp .OhiqDr.D.^rD^q: 

[Syll] D h :.p. D\ qDr : D : p . D . ~r D ~^ 

h.Exp. D r- z.p.q. D .r : D :^>. D . # Dr 

h .Imp . DF:.^).D.'>JrD~g , :^:p.~r.D.~g , 

h . (2) . (1) . (3) . Syll . D K Prop 



(1) 
(2) 
(3) 



SECTION A] THE LOGICAL PEOGDUCT OF TWO PROPOSITIONS 113 

This is another form of transposition. 
*3*4. H : p . q . D . p D q [*251 . Transp . (*1'01 . *301)] 

*3'41. bz.pDr.Dzp.q.O.r [*3-26.Syll] 
*342. \-z.q3r.Dzp.q.D.r [*327.Syll] 
*343. t-z.pDq.pDr.Dzp.O.q.r 

Dem. 

K*3 > 2.DH:.?.D:r.D.g.r (1) 

\-.(l).Syll.D\-zzpDq.Dz.p.Dzr.D.q.rz. 

[*2-77] Oz.pDr.Dzp.D.q.r (2) 

f- . (2) . Imp . 3 > . Prop 

*344. b z.qDp.rDp.Dzqvr .D .p 

This principle is analogous to #343. The analogy between *3'43 and 
#3*44 is of a sort which generally subsists between formulae concerning 
products and formulae concerning sums. 

Dem. 

h . Syll . 3 H :. ~ q 3 r . r Dp . 3 : ~ q 3 p z 

[*2*6] DzqDp.D.p (1) 

I- . (1) . Exp . 3 f- :: ~ q 3 r . 3 :. r 3p . 3 : <j 3p . 3 . p :. 

[Comm.Imp] 3 :. # Dp . r 3 p . 3 . p (2) 

I- . (2) . Comm . 3 h :. q Dp . r 3 p . 3 : ~ q 3 r . 3 .p :. 

[*2'53.Syll] 3 h . Prop 

#3*45. I- :.p3<j.3 :p.r .3.g.r 

This principle shows that we may multiply both sides of an implication 
by a common factor; hence it is called by Peano the "principle of the factor." 
We shall refer to it as " Fact." It is the analogue, for multiplication, of the 
primitive proposition #1*6. 

Dem. 

I- .Syll . 3 h z.pDq. 3:gO~r.3.p3~r: 

[Transp] 3 : ~(p 3~r) . 3 . ~(gO~r) :. 

[Id.(*l-01.*3-01)] 3 h . Prop 

*3"47. b z.pDr . qDs. 3 \p .q. 3.r .s 

This proposition, or rather its analogue for classes, was proved by Leibniz, 
and evidently pleased him, since he calls it "praeclarum theorema*." 

Dem. 

K*3-26.3h:.p3r.£3s.3:p3r: 

[Fact] Dzp.q-.D.r.qz 

[*3-22] Dzp.q.D.q.r (1) 

* Philosophical works, Gerhardt's edition, Vol. vii. p. 223. 
R&w I 8 



114 MATHEMATICAL LOGIC [PABT I 

K #3-27. D \-:.p Dr. q Ds.DzqOs: 
[Fact] D:g.r.O.*.r: 

[*3'22] D-.q.r.O.r.s (2) 

K(1).(2).*303.*2-83.D 

I- :.p D r . ? D s . D : p . q ..? . r . s :. D H . Prop 
*348. h:.^Dr.grDs.D:j>v^.D.rvs 
This theorem is the analogue of *3'47. 

Bern. 

V . #3-26 . D f- :.p D r . q D s, D : p 3 r : 

[Sum] Dipvg.D.rv^: 

[Perm] Ozpvq.O.qvr (1) 

I- . *3-27 . D I- :.£> Dr . q D « . D : q Ds : 

[Sum] Dztjvr.D.svr: 

[Perm] Dr^vr.D.rvs (2) 

H.(1).(2).*2-83.D 

I- :. p D r . # D s . D :p v <f . D . r v s :. D h . Prop 



*4. EQUIVALENCE AND FORMAL RULES 

Summary of #4. 

In this number, we shall be concerned with rules analogous, more or less, 
to those of ordinary algebra. It is from these rules that the usual " calculus 
of formal logic " starts. Treated as a " calculus," the rules of deduction are 
capable of many other interpretations. But all other interpretations depend 
upon the one here considered, since in all of them we deduce consequences 
from our rules, and thus presuppose the theory of deduction. One very 
simple interpretation of the " calculus " is as follows : The entities considered 
are to be numbers which are all either or 1 ; " p D q" is to have the value 
if p is 1 and q is ; otherwise it is to have the value 1 ; ~p is to be 1 if p 
is 0, and if p is 1 ; p . q is to be 1 if p and q are both 1, and is to be in 
any other case ; p v q is to be if p and q are both 0, and is to be 1 in any 
other case; and the assertion-sign is to mean that what follows has the 
value 1. Symbolic logic considered as a calculus has undoubtedly much 
interest on its own account ; but in our opinion this aspect has hitherto been 
too much emphasized, at the expense of the aspect in which symbolic logic 
is merely the most elementary part of mathematics, and the logical pre- 
requisite of all the rest. For this reason, we shall only deal briefly with what 
is required for the algebra of symbolic logic. 

When each of two propositions implies the other, we say that the two are 
equivalent, which we write " p = q." We put 

*401. p=q. = .pDq.q0p Df 

It is obvious that two propositions are equivalent when, and only when, 
both are true or both are false. Following Frege, we shall call the truth- 
value of a proposition truth if it is true, and falsehood if it is false. Thus two 
propositions are equivalent when they have the same truth-value. 

It should be observed that, if p = q, q may be substituted for p without 

altering the truth-value of any function of p which involves no primitive 

ideas except those enumerated in *1. This can be proved in each separate 

case, but not generally, because we have no means of specifying (with our 

apparatus of primitive ideas) that a function is one which can be built up out 

of these ideas alone. We shall give the name of a truth-function to a function 

f(p) whose argument is a proposition, and whose truth- value depends only 

upon the truth- value of its argument. All the functions of propositions with 

which we shall be specially concerned will be truth-functions, i.e. we shall 

have 

P = <1- 3- /(P) =/(<!)■ 

8—2 



116 MATHEMATICAL LOGIC [PART I 

The reason of this is, that the functions of propositions with which we deal 
are all built up by means of the primitive ideas of #1. But it is not a universal 
characteristic of functions of propositions to be truth-functions. For example, 
"A believes p* may be true for one true value of p and false for another. 

The principal propositions of this number are the following: 
*4"1. h zp'Dq. = . ~gD~p 
#4*11. I- :p = q. = .<>jp = ~q 

These are both forms of the " principle of transposition." 
#4:13. h.^ = ~(~#) 

This is the principle of double negation, i.e. a proposition is equivalent to 
the falsehood of its negation. 
#42. \-.p=p 
*421. h :p = q.=.q=p 
#422. H :p~q.q = r . D .p = r 

These propositions assert that equivalence is reflexive, symmetrical and 
transitive. 

#424. h :p.= .p .p 
#425. h:p. = .pvp 

I.e. p is equivalent to "p and p" and to " p or p" which* are two forms of 
the law of tautology, and are the source of the principal differences between 
the algebra of symbolic logic and ordinary algebra. 
#4'3. h:p.q.=.q.p 

This is the commutative law for the product of propositions. 
#431. h:pvq. = .qvp 

This is the commutative law for the sum of propositions. 

The associative laws for multiplication and addition of propositions, namely 
#4 - 32. h : (p . q) . r . = . p . (q . r) 
#433. h :(p vq) vr . = .p v(g-vr) 

The distributive law in the two forms 
#44. \-:.p.qvr. = :p.q.v.p.r 
#441. h :.p . v.q . r : = .p vq .pvr 

The second of these forms has no analogue in ordinary algebra. 
#4*71. h :.pDq.~:p . = .p .q 

I.e. p implies q when, and only when, p is equivalent to p . q. This pro- 
position is used constantly; it enables us to replace any implication by an 
equivalence. 
#4'73. h :. q . D :p . = .p . q 

I.e. a true factor may be dropped from or added to a proposition without 
altering the truth-value of the proposition. 



SECTION A] EQUIVALENCE AND FORMAL RULES 

#401. p = q* = .pOq.q Op Df 
*402. p = q = r.=z.p = q.q = r Df 

This definition serves merely to provide a convenient abbreviation. 



117 



•41. 


H :^>Dg. = m ojqD~p 






[•2-16-17] 


•411. 


\- :p = q. = .f*p = ~q 






[•216-1 7. •3-47-22] 


•412. 


h zp = ~q m = ,q = r*>p 






[•20315] 


•413. 


\r . p = <v* (~jp) 






[•2-12-14] 


•414. 


h i.p.q. D.r : = :_p.~r< 


>. 


~g 


[*337. *413] 


•415. 


h z.p.g.D.^r: = :g.r 


■ 3. 


<vj) 


[#3-22 . •41314] 


•4*2. 


H.^)=^ 






[Id.*32] 


•4*21. 


h:p = ?. = .?=;> 






[*322] 


•4*22. 


|-:^ = gf.g = r.D.p = r 









(1) 

(2) 
(3) 

(4) 

(5) 
(6) 



Dem. 

h.*3*26. ~2t-:p = q.q = r.'D.p = q. 

[•3-26] i-piq 

K*3*27. 2\-:p = q.q = r.'2.q==r. 

[•3-26] D.gDr 

h . (1) . (2) . #2*83 .3\-:p = q.q = r.3.pDr 
h . *3'27 . Db:p-=q.q = r.0.q = r. 

[*3'27] D.rOq 

K*326. 3bip = q .q = r.0.p = q. 

[•3-27] 3-qOp 

b . (4) . (5) . *2'83 .0\-:p = q.q = r.0.rOp 
K(3).(6).Comp.Dh.Prop 

Note. The above three propositions show that the relation of equivalence 
is reflexive (•4*2), symmetrical (•4*21), and transitive (•4*22). Implication 
is reflexive and transitive, but not symmetrical. The properties of being 
symmetrical, transitive, and (at least within a certain field) reflexive are 
essential to any relation which is to have the formal characters of equality. 

*4'24. \-:p. = .p.p 

Bern. 

\-.*B-26.Db:p.p.D.p (1) 

h.*3-2. DH.jp. D zp.D.p.p:. 

[•2-43] Db-.p.O.p.p (2) 

I- . (1) . (2) . *3-2 . K Prop 



*4 25. h:p.z=.pvp Taut . Add ^ 



Note. *4'24'25 are two forms of the law of tautology, which is what chiefly 
distinguishes the algebra of symbolic logic from ordinary algebra. 



118 MATHEMATICAL LOGIC [PART I 

*4-3. \-:p.q. = .q.p [*3'22] 

Note. Whenever we have, whatever values p and q may have, 

<f>(p,q).y.<f>(q,p), 
we have also 

<f>(p,q).= .<f>(q,p). 

For {* (p,q).D.<f> (q,p)} %* . D : <j> (q,p) . D . tf> (p, <?). 

*431. H i^vg . = . qvp [Perm] 

*4'32. H : (p . g) . r . = . p . (# . r) 

Dem. 

K#4 # 15. D\-:.p.q.D.~r: = :q.r.'2.~p: 

[*4-12] =:2>.D.~(gr.r) (1) 

h . (1) . *4'11 . D h : ~ (p . 5 . D . ~ r) . = . ~ ( j) . D . ~ (^ . r)} : 
[(*r01.*301)] D h . Prop 
jVofe. Here "(1)" stands for " H :.p . q. D . ~r : = :jp . 3 . ~ (<? . r)," which 

is obtained from the above steps by #4*22. The use of #4*22 will often be 

tacit, as above. The principle is the same as that explained in respect of 

implication in #231. 

*4-33. \-:(pvq)yr. = .pv(qvr) [*231'32] 

The above are the associative laws for multiplication and addition. To 

avoid brackets, we introduce the following definition : 

*4*34. p.q.r . = .(p.q).r Df 

*4'36. h:.p = q.D:p . r.=.q.r [Faet.*3-47] 

*437. \-:.p = q.D:pvr. = .qvr [Sum.*3-47] 

*438. \-:.p=r.q=s.D:p.q. = .r.s [*3'47 . *432 . *3-22] 

*4-39. V'..p = r.q=s.^:p\q. = .rys [*3*48-47 . *432 . *3'22] 

*4*4. h z.p.qvr . = :p.q . v .p .r 

This is the first form of the distributive law. 

Bern. 

K*3'2. 0\-::p.D:q.'2.p.qz.p.'2:r 

[Comp] 0\-::p.D:.q.y.p.q:r.D.p, 

[*3'48] Dz.qvr. D-.p.q.v.p. 

f-.(l).Imp. 0\-:.p.qvr.D:p.q. v,p.r 

K#3"26. D\-:.p.q.D.p:p.r.D.p:. 

[*3'44] Dhz.p.q. v .p.r-.D.p (3) 

h.*3'27. D\-:.p.q.y.q:p.r.'D.r:. 

[*3-48] yli.p.q.v .p.r-'.y.qvr (4) 

h . (3) . (4) . Comp . Df :.p.q. v .p.rzD.p. qvr (5) 

r.(2).(5). DKProp 



D, 


■P- 


, r ::. 




r :, 








r 






(1) 
(2) 



SECTION A] EQUIVALENCE AND FOEMAL BULES 119 

*4'41. \-:.p.v .q.r:= .pvq.pvr 

This is the second form of the distributive law— a form to which there 
is nothing analogous in ordinary algebra. By the conventions as to dots, 
"p . v . q . r n means "p v (q . r)." 

Dem. 

h . *3'26 . Sum . D h :.p. v. q.r : D .p vq (1) 

I- . *3'27 . Sum . D r- up . v . q . r : D .p v r (2) 

K(l).(2).Conip.Dh:.jp.v.g.r:D._pvgr.jpvr (3) 

h . *2*53 . *3*47 . DH:._pvg'.^vr.3:~|)Dg'.~i)3r: 
[Comp] D:~p.D.q.r: 

[*2'54] D:p.v.g.r (4) 

K(3).(4). DKProp 

*4*42. f- :.^>. = zjj.g. v.^.^^ 

Ztewi. 

K*3'21 ... Dh:.5V~g.D:^.3.j5.^v~g:. 

[*211] Or-rp.D.^.grv^ (1) 

h.*3-26. Dh:j).grv~g. j|.|> (2) 

H . ( 1 ) . (2 ) .O I- : . p . = : p . q v ~ q : 
[*4*4] =:p.q.v .p.~q:.Dh . Prop 

*4'43. h:.p. = :pvq.pv~q 

Dem. 

t-.*2*2. Dhip.D .pvqzp.D.pv^q: 

[Comp] Dbip.'Z.pvq.pv^q (1) 

b . *265 ^ . D I- :. ~p D g . D : ~p 3 ~» o . D .p :. 
[Imp] Dh :.~j)Dgf.~^D~gr .D.p:. 

[*2-53.*3*47] Dhz.pv^.pv^^.D.^ (2) 

K(l).(2). Oh. Prop 

#444. \-:.p. = :p.v.p.q 

Dem. 

h . *2'2 . ^Vi.p.^-.p.y.p.q (1) 

h . Id . *3'26 . D V :.p "Dp :p'. q . O .p :. 
[*3'44] Dbt.p.v.p.qiD.p (2) 

K(l).(2). DKProp 

*445. I- : p . = . p . J? v q [*3'26 . *2*2] 

The following formulae are due to De Morgan, or rather, are the propo- 
sitional analogues of formulae given by De Morgan for classes. The first 
of them, it will be observed, merely embodies our definition of the logical 
product. 



120 



MATHEMATICAL LOGIC 



[PART I 



*4'5. 
*451. 


h 




*452. 


> 


! p . <**>q . 


*453. 


f- 


: *^» (p . <^» q) . 


*454. 


r. 


~p. q. 


*4-55 ; 


h: 


o->(cvp . £). 


*456. 


I-: 


<^J£> .<vj. 


*457. 


h 


:^>(<^>jp.no«). 



= .~(^v<v ? ) [*4-2 . (*3-01)] 
= .<^>j)V~g' [*4*512] 



= .~(~pvq) 

= . ^ (p v «m q) 
= . p v «v» q 
= .~(pvq) 
= .pvq 

The following formulae are obtained immediately from the above. They 
are important as showing how to transform implications into sums or into 
denials of products, and vice versa. It will be observed that the first of them 
merely embodies the definition *l a 01. 



r*4.5^.* 4 .i3"| 
04-52-12] 
f*4-5^.*413] 
[*4'5412] 

[*4-5612] 



*4'6. 1- : 


p3q.= . r^pwq 


[*4-2 . (*1-01)] 


*461. h: 


~(pDq).=.p.~q 


[«4'6'll-52] 


*4'62. H 


p D~ q . =. ~|)v<v q 


H t] 


*463. r: 


~(pD~q) , = .p.q 


[♦4-6211-5] 


*464. h: 


r^pDq . = .pv q 


[*2'53'54] 


*4/65. h: 


(**> (<~J3 D q) . = . r*> p . <>>> q 


[*4-6411-56] 


*466. h: 


o->p^e>jq. = .pvt**>q 


[•4-64 ^2] 


*467. h: 


<**> (~p 3 ~ q) . = . ~p • 9 


[*4-66-ll'54] 


*47. h : 


.pDq.= :p .0 .p.q 




Bern. 







h.*3-21.Sy\l.D\-:.p.D.p.q:0.p3q 
KComp. O^z.pOp.pOq.Dzp.'D.p.q:,. 

[Exp] D\-::pDp.D:.pOq.D:p.D.p.q:: 

[Id] Dh i.pDq.'Dzp.D .p.q 

h.(l).(2). DKProp 

pDq. 



(1) 

(2) 



:p.=.p.q 



*471. > 

Dew. 

K*3-21. Dhi-.p.q.D.p-.D 

[*3'26] Dh:.p.D.jp. ? :D 

K*3'26.. 3h:.p. = .p.q:3 

K(l).(2). 3r-:.f».D..p.g:» 
K (3) . *4-7-22 . D h . Prop 



p. = .p.q (1) 

p.Z.p.q (2) 

P- = -2>-3 (3) 



SECTION A] EQUIVALENCE AND FORMAL RULES 121 

The above proposition is constantly used. It enables us to transform 
every implication into an equivalence, which is an advantage if we wish to 
assimilate symbolic logic as far as possible to ordinary algebra. But when 
symbolic logic is regarded as an instrument of proof, we need implications, 
and it is usually inconvenient to substitute equivalences. Similar remarks 
apply to the following proposition. 

*4'72. \-:.pOq.= :q. = .pvq 

Dem. 

\- . #41 . D b :. p D q . = : ~ q D ~p : 

#4*71 "^-'-"^l =:~o. = .~g.~p: 
■ 1 P> <l J 

[*4'12] =:q. = .~>(~q m r»p): 

[*4'57] = : q . = . q Vp : 

[#4*31] ==:q. = .pvq:.D\-. Prop 

#473. \-:.q.'5ip. = .p.q [Simp. #4-71] 

This proposition is very useful, since it shows that a true factor may be 
omitted from a product without altering its truth or falsehood, just as a true 
hypothesis may be omitted from an implication. 

#474. \-:.~p.Dzq. = .pvq [#2-21 . #472] 



#441 ^.(#101)1 



#476. \-:.pDq.p'2r.= :p.'D.q.r 

#477. b:.qOp.rOp. = :qyr.0.p [*344 . Add . *2'2] 

#478. \-:.pOq.v .pOr: = :p.D .qvr 
Dem. 
\- .#4*2 .(#101). Dr-:.jOgr.v.jOr: = :<>->pvq.v .~pvr; 
[#4*33] = . <^jp . v . q v^p vr : 

[#4'31*37] = : e>jp. v . ^p vq vr : 

[#4*33] =:<v^v~j).v.g , vr: 

[*4*25'37] = : ~p . v . q v r : 

[*4-2.(*101)] =:p.D.gvr:.DI-.Prop 

#479. bz.q'Dp. v.r'Dp: = :q.r . D.p 
Dem. 
K*4 , l*39. D h i.qDp.v .r"2p : = : ~pD~q . v.~pD~r : 
[#4*78] E:~j).0.~5v~r: 

[#2 - 15] = z™(<*^qv ™r) .D .p: 

[*4'2.(#301)] =:q.r.D.p:.D\-.?rov 

Note. The analogues, for classes, of #47879 are false. Take, e.g. #4-78, 
and put p = English people, q — men, r — women. Then p is contained in q 
or r, but is not contained in q and is not contained in r. 



122 



MATHEMATICAL LOGIC 



[PART 1 



*4r*8. l-:pD~p. = .~|> [*2'01 . Simp] 

#481. \-:^pDp. = .p [*2'18.Simp] 

*4 82. h r p D-g . p 3 ~ q . ; = . ~p [*2'65 . Imp . *2'21 . Comp] 

*483L \-:pOq.r^p~yq. = .q [#2'61 . Imp . Simp . Comp] 

Note. *4'82*83 may also be obtained from *4-43, of Tyhich they are virtu- 
ally other forma 

*4'84 \-'..p=q.^:pyr. = .qDr [*2*()6 . *3'47] 
*485. h:.p = q.yzrDp.= .rOq [*205 . *3'47] 
*486. \-up=q.^:p = r. = .q = r [*4'21-22] 

*487. I" up . q . D . r : , = : p .3 .qD r : = : q . 3 . jO r:s: £ . p . D . r 

[Exp . Comm . Imp] 

#4*87 embodies in one proposition the principles of exportation and im- 
portation and the commutative principle. 



*5. MISCELLANEOUS PROPOSITIONS 

Summary of #5. 

The present number consists chiefly of propositions of two sorts: (1) those 
which will be required as lemmas in one or more subsequent proofs, (2) those 
which are on their own account illustrative, or would be important in other 
developments than those that we wish to make. A few of the propositions of 
this number, however, will be used very frequently. These are : 
*5*1* I- : p . q . D . p = q 

I.e. two propositions are equivalent if they are both true. (The statement 
that two propositions are equivalent if .they are both false is #5 '21.) 

*5*32. I- :.p . D . q = r := : p.q. = .p.r 

I.e. to say that, on the hypothesis p, grand r are equivalent, is equivalent 
to saying that the joint assertion of p and q is equivalent to the joint assertion 
of p and r. This is a very useful rule in inference. 

#56. hz.p.f^q.D .r : = :p. >.#vr 

I.e. "p and not-q imply r" is equivalent to "p implies q or r." 

Among propositions never subsequently referred to, but inserted for their 
intrinsic interest, are. the following: #5*1I , 12 , 13*I4, which state that, given 
any two propositions p, q, either p or ™>p must imply q, and p must imply 
either q or not-*?, and either p implies g or q implies p; and given any third 
proposition r, either p implies q or q implies r*. 

Other propositions not subsequently referred to are *5*22*23'24; in these 
it is shown that two propositions are not equivalent when, and only when, 
one is true and the other false, and that two propositions are equivalent 
when, and only when, both are true or both false. It follows (#5*24) that the 
negation of "p . q . v . ~p . ~ q" is equivalent to "p .^q.v.q. <^p." #5"54'55 
state that both the product and the sum of p and q are equivalent, respectively, 
either to p or to q. 

The proofs of the following propositions are all easy, and we shall therefore 
often merely indicate the propositions used in the proofs. 



#6'1. H : p. q, D . p = q JW4-22] 

#511. h zpOq.v .r-^pDq [*2-5'54] 

#512. \-zpDq.v .pD~q [*2-51*54] 

#513. h-.p^q.v.qDp [*2'521] 

#514. H : p Dq . v . qDr [Simp . Transp . #2*21] 

* Cf. Schroder, Vorlesungen liber Algebra der Logik, Zweiter Band (Leipzig, 1891), pp. 270— 
271, where the apparent oddity of the above proposition is explained. 



124 



MATHEMATICAL LOGIC 



[PART 1 



#5*15. h:p = q.v.p = ~q 

Bern. 

V . *4'61 . D H : ~ (p D q) . D .p . ~ q . 

[*51] D.p = ~q: 

[*2-54] Dh:jpDgr.v.p = ~5 (1) 

h , #461 . D h : ~ (y D j>) . D . q . ~p . 

[#51] D . 9 = ~p . 

[#412] 0.p=~r>jq; 

[*2 54] Dhs^Dp.v.jpi^g' (2) 

h.(I).(2).#4'41.DKProp 

#516. h . ~ (p = q . p = ~ q) 

Bern. 

b .*326. Oh zp = q.pD~q.O .pOq.pO ~q- 

[*4'82] D.~p (1) 

I- . *3'27 . D h :p = q .p D ~ q . D . q Op . /> D ~ q . 

[Syll] D. 3 D~0. 

[Abs] D.~? (2) 

I- . (1) . (2) . Comp . D \- : p = q . p D ~ q . 3 . ~p . ~<? . 

f*4-65^ D.~(~ 2 :>p) (3) 

K (3) . Exp . D h :. p = q . D : p D ~? . D . ~(~2 Dp) : 
[Id.(*101)] 3:~(/0~g).v.~(~2:>2>): 

[*4'51 .(#401)] D : ~ {p = ~ q) : . D h . Prop 

#517. H : jpvgr. ~(jp . g) . = ,p = ~q 

Dem. 

K*4'64'21. Ohipvq . = .~qOp (1) 

h . #463 . Transp . }\- ;~(p .q). = .pO~q (2) 

h . (1) . (2) . *4-38-21 . D H . Prop 



#518. h:p = q. = . ~ (p = ~ q) 



[♦51516 ■«6iy p5g ' 1,5 "' g l 
|*518 ^.*4- 



#519. K~(p = ~p) 

*5'21. \-:~p.~q.0.p = q [*51 . #411] 

#522. h:.~(p = q). = :p.~q.v.q.~p [#4'61-51'39] 

#5-23. \-:.p = q. = :p.q.v.~p.~q |*518 .*5'22^ .*413-36 

#5'24. h:.~(p.q.v m ~>p.~q). = :p.~q.v.q.~p [#5'22-23] 

#5-25. b:.pvq. = :pOq.D.q [*2-62'68] 



SECTION A] MISCELLANEOUS PROPOSITIONS 125 

From *5*25 it appears that we might have taken implication, instead of 
disjunction, as a primitive idea, and have defined "pvq" as meaning 
"pDq.D . q." This course, however, requires more primitive propositions 
than are required by the method we have adopted. 

*53. \-:.p.q.0.r: = :p.q.'2.p.r [Simp*Comp»Sy)l] 
*5'31. b:.r.p^q'.0:p.D.q.r [Simp.Comp] 

*5-32. b:.p.3.q = r: = :p.q. = .p.r [*4'76 . *3331 . *53] 
This proposition is constantly required in subsequent proofs. 



*533. 


b ;* p . qD r . = : p : p . q . D . r 




[*4-73-84 


. *5-32] 


*535. 


b:.pDq.pDr.^:p.D.q = r 




[Comp . *5*1] 


*5 36. 


b:p.p=q. = .q.p = q 




[Ass . *4-38] 


*54. 


b:.p.D.pDq: = .pDq 




[Simp.*2-43] 


*541. 


bz.pDq.D ,p'Dr:=ip.7J.q'2r 




[*2-77-8G] 


*542. 


b ::p . D . q"5r: = :,p . D :q. O .p 


.r 


[*5-3 . *4-87] 


*544 


b :: p D q . D :,p D r . = : p . D . q . r 




[*4-76 . *53-32] 


*5*5. 


b :.p. D : jOg. = .q 




[Ass . Exp . Simp] 


*5 501 


.h:.p\3:q. = .p = q 




[*5*1 . Exp . Ass] 


*5'53. 


b:.pvqvr.'2.s: = :p'Ds.q'2s. 


rDi 


• [*4-77] 


*5 54. 


b :. p . q . = . p : v : p . q . = . q 


[*4- 


73 . *444 . Transp . *51] 


*5 55. 


b: m pvq. = .p;v:pvq. = .q 


[*1* 


3 . *51 . *4-74] 


*56. 


b:.p.~q.D.rz = :p.'5.qvr 


1*4-87 -^ . *4-648ol 


*561. 


b : p v q . <^q . = ,p . ~q 


[*4* 


74.*532] 


*5 62. 


b :.p .q.v .~q : = .pv~q 


f *4-7 ^~ 




*563. 


b \.pv q. = ip .v .cop . q 


[*5-62~^1 

L q> p\ 




*57. 


b :.pv7' . = .qvr : = ir.v.p^q 


[*4 


74.*l-3.*51.*4-37] 


*571. 


b :. qD~r.'D:pyrq.r. = .p.r 











In the following proof, as always henceforth, "Hp" means the hypothesis 
of the proposition to be proved. 

Dem. 

h.*4-4. Db:.pvq.r.= :p.r.v.q.r (1) 

V . *4-62ol .O b :: Hp . D :. ~ (q . r) :. 

[*4'74] D:.p.r.v.q.r: = :p.r (2) 

b .(1) . (2) . *4-22 . D b . Prop 



126 MATHEMATICAL LOGIC [PART I 

*5*74. f :. p . y.q~=r : = ^ j) 3 j...~ . j)D r 
Dem. 
b . #5*41 . D h r:j> D q . D .£> D r : = :p . >. g D r :. 

pDr.D.pDg 1 : = :p. D.rOg (1) 

r- . (1) . *4'38 . D h ::p D g . s . jO r . .= :..p . D .ql>r:p. D . rO^ :. 
[#4"76] = :. jp . D . 3 = r :: D I- .Prop 

*5'75. b:.r'D<^>qip. = .qvr:D:p.<^q.= .r 

Dem. 

h.*5-6. Df-:.Hp.D:p.~^.D.r (1) 

I- . *327 . D h :. Hp . D : g- v r . D .p : 

[*4-77] D:rDp (2) 

h.*3-26.Dh:.Hp.D:rD~2 (3) 

f- . (2) . (3) . Comp . D h :. Hp . D : r Dp . r D ~ q : 

[Comp] 3 : r . D . jp . ~ q (4) 

H . (1) . (4) . Camp . D b :. Hp .D:|).~g. = .r:.Dh. Prop 



SECTION B 
THEORY OF APPARENT VARIABLES 

*9. EXTENSION OF THE THEORY OF DEDUCTION FROM 
LOWER TO HIGHER TYPES OF PROPOSITIONS 

Swnmary of*9. 

In the present number, we introduce two new primitive ideas, which may- 
be expressed as "<f>x is always* true" and "<f>x is sometimes* true," or, more 
correctly, as "<j>x always" and "<fix sometimes." When we assert "0# always," 
we are asserting all values of <£&, where "<p$" means the function itself, as 
opposed to an ambiguous value of the function (cf. pp. 15, 40); we are not 
asserting that <f>x is true for all values of x, because, in accordance with the 
theory of types, there are values of x for which "<f>x" is meaningless; for ex- 
ample, the function $& itself must be such a value. We shall denote "$x 
always" by the notation 

(x) . $x, 

where the "(x)" will be followed by a sufficiently large number of dots to 
cover the function of which "all values" are concerned. The form in which 
such propositions most frequently occur is the "formal implication," i.e. such 
a proposition as 

(x) : <f>Xm y.tyx, 

i.e. "<f>x always implies yfrx." This is the form in which we express the 
universal affirmative "all objects having the property £ have the propertyi/r." 
We shall denote "<f>x sometimes" by the notation 

fax).<f>x. 
Here "g" stands for "there exists," and the whole symbol may be read 
"there exists an x such that <j>x." 

In a proposition of either of the two forms (x) . fa, fax) . fyx, the x is 
called an apparent variable. A proposition which contains no apparent 
variables is called "elementary," and a function, all whose values are elemen- 
tary propositions, is called an elementary function. For reasons explained in 
Chapter II of the Introduction, it would seem that negation and disjunction 
and their derivatives must have a different meaning when applied to elemen- 
tary propositions from that which they have when applied to such propositions 
as (x) . <j>x or fax) . $x. If $x is an elementary function, we will in this number 
call (x) . <f>x and fax) . <f>x "first-order propositions." Then in virtue of the fact 

* We use "always" as meaning "in all eases," not "at all times." A similar remark applies 
to "sometimes." 



128 MATHEMATICAL LOGIC [PABT I 

that disjunction and negation do not have the same meanings as applied to 
elementary or to first-order propositions, it follows that, in asserting the 
primitive propositions of #1, we must either confine them, in their application, 
to propositions of a single type, or we must regard them as the simultaneous 
assertion of a number of different primitive propositions, corresponding to the 
different meanings of "disjunction" and "negation." Likewise in regard to 
the primitive ideas of disjunction and negation, we must either, in the primi- 
tive propositions of *1, confine them to disjunctions and negations of elementary 
propositions, or we must regard them as really each multiple, so that in regard 
to each type of propositions we shall need a new primitive idea of negation 
and a new primitive idea of disjunction. In the present number, we shall 
show how, when the primitive ideas of negation and disjunction are restricted 
to elementary propositions, and the p, q, r of *1 — *5 are therefore necessarily 
elementary propositions, it is possible to obtain definitions of the negation and 
disjunction of first-order propositions, and proofs of the analogues, for first- 
order propositions, of the primitive propositions *1*2 — 6. (*11 and #1*11 
have to be assumed afresh for first-order propositions, and the analogues of 
*l-7-71 i 72 require a fresh treatment.) It follows that the analogues of the 
propositions of *2— *5 follow by merely repeating previous proofs. It follows 
also that the theory of deduction can be extended from first-order propositions 
to such as contain two apparent variables, by merely repeating the process 
which extends the theory of deduction from elementary to first-order pro- 
positions. Thus by merely repeating the process set forth in the present 
number, propositions of any order can be reached. Hence negation and 
disjunction may be treated in practice as if there were no difference in these 
ideas as applied to different types^Nthat is to say, when " ~ p" or "pvq" 
occurs, it is unnecessary in practice to know what is the type of p or q, since 
the properties of negation and disjunction assumed in *1 (which are alone used 
in proving other properties) can be asserted, without formal change, of pro- 
positions of any order or, in the case of p v q, of any two orders. The limitation, 
in practice, to the treatment of negation or disjunction as single ideas, the 
same in all types, would only arise if we ever wished to assume that there is 
some one function of p whose value is always ~ p, whatever may be the order 
of p, or that there is some one function of p and q whose value is always p vq, 
whatever may be the orders of p and q. Such an assumption is not involved 
so long as p (and q) remain real variables, since, in that case, there is no need 
to give the same meaning to negation and disjunction for different values of 
p (and q), when these different values are of different types. But if p (or q) 
is going to be turned into an apparent variable, then since our two primitive 
ideas (x) . 4>x and (g#) . <f>x both demand some definite function <f>, and restrict 
the apparent variable to possible arguments for </>, it follows that negation 
and disjunction must, wherever they occur in the expression in which p (or q) 
is an apparent variable, be restricted to the kind of negation or disjunction 



SECTION B] EXTENSION OP THE THEORY OF DEDUCTION 129 

appropriate to a given type or pair of types. Thus, to take an instance, if we 
assert the law of excluded middle in the form. 

"\-.pv~p" 
there is no need to place any restriction upon p: we may give to p a value 
of any order, and then give to the negation and disjunction involved those 
meanings which are appropriate to that order. But if we assert 

" ^ • (p) -pv ^p" 
it is necessary, if our symbol is to be significant; that "p v ~ p" should be the 
value, for the argument p, of a function <fyp\ and this is only possible if the 
negation and disjunction involved have meanings fixed in advance, and if, there- 
fore, p is limited to one type. Thus the assertion of the law of excluded middle 
in the form involving a real variable is more general than in the form involving 
an apparent variable. Similar remarks apply generally where the variable is 
the argument to a typically ambiguous function. 

In what follows the single letters p and q will represent elementary pro- 
positions, and so will "<£#," "yfrx," etc. We shall show how, assuming the 
primitive ideas and propositions of #1 as applied to elementary propositions, 
we can define and prove analogous ideas and propositions as applied to pro- 
positions of the forms (x) . <f>x and (<^x).<f>x. By mere repetition of the analogous 
process, it will then follow that analogous ideas and propositions can be defined 
and proved for propositions of any order; whence, further, it follows that, in 
all that concerns disjunction and negation, so long as propositions do not 
appear as apparent variables, we may wholly ignore the distinction between 
different types of propositions and between different meanings of negation 
and disjunction. Since we never have occasion, in practice, to consider pro- 
positions as apparent variables, it follows that the hierarchy of propositions 
(as opposed to the hierarchy of functions) will never be relevant in practice 
after the present number. 

The purpose and interest of the present number are purely philosophical, 
namely to show how, by means of certain primitive propositions, we can 
deduce the theory of deduction for propositions containing apparent variables 
from the theory of deduction for elementary propositions. From the purely 
technical point of view, the distinction between elementary and other propo- 
sitions may be ignored, so long as propositions do not appear as apparent 
variables; we may then regard the primitive propositions of #1 as applying 
to propositions of any type, and proceed as in #10, where the purely technical 
development is resumed. 

It should be observed that although, in the present number, we prove 
that the analogues of the primitive propositions of #1, if they hold for propo- 
sitions containing n apparent variables, also hold for such as contain n + 1, 
yet we must not suppose that mathematical induction may be used to infer 
that the analogues of the primitive propositions of #1 hold for propositions 

r& w i 9 



130 MATHEMATICAL LOGIC [PAKT I 

containing any number of apparent variables. Mathematical induction is a 
method of proof which is not yet applicable, and is (as will appear) incapable 
of being used freely until the theory of propositions containing apparent 
variables has been established. What we are enabled to do, by means of the 
propositions in the present number, is to prove our desired result for any as- 
signed number of apparent variables — say ten^ — by ten applications of the same 
proof. Thus we can prove, concerning any assigned proposition, that it obeys 
the analogues of the primitive propositions of #1, but we can only do this by 
proceeding step by step, not by any such compendious method as mathematical 
induction would' afford. The fact that higher types can only be reached step 
by step is essential, since to proceed otherwise we should need an apparent 
variable which would wander from type to type, which would contradict the 
principle upon which types are built up. 



Definition of Negation. We have first to define the negations of (x) . <j>x 
and (go?) . <f>x. We define the negation of (x) . <f>% as fax) . ~ 4>x, i.e. "it is 
not the case that <f>x is always true" is to mean "it is the case that not-$# 
is sometimes true." Similarly the negation of (gar) . <f>x is to be defined as 
{x) . ~ <j>x. Thus we put 
*901. ~ {(x) . <f>x} . = . (gar) . ~ <f>x Df 
*902. ~> {fax) . <f>x} . = .(#). ~ <f>x Df 

To avoid brackets, we shall write ~ (x) . <f>x in place of ~ {(x) . <f>x} y and 
-v fax) . <px in place of ~ {(3#) ■ 4*®]- Thus : 
*9011. ~ (x) . <\>x . = . ~ {(x) . (f>x} Df 

*9 021. ~ fax) . <f>x . = . ~ {fax) . <f>x\ Df 

Definition of Disjunction. To define disjunction when one or both of the 
propositions concerned is of the first order, we have to distinguish six cases, 
as follows: 

*903. (x). <f>x . v . j> :=.(#) . <f>x vp Df 
#904. p . v . (x) . <£# : = . (x) . p v <f>x Df 
#9*05. fax) -. (j>x . v . p : = . fax) . <f>x vp Df 
*9*06. p . v . fax) ,<f)xz = . fax) .pv<px Df 
*9'07. (x) . <f>x . v . fay) . yfry: = : (x) : fay) .fyxytyy Df 
*908. fay) . yjry . v . (a?) . <f>x : = : (x) : fay) . yfry v <f>x Df 

(The definitions #907'08 are to apply also when <f> and i/r are not both 
elementary functions.) 

In virtue of these definitions, the true scope of an apparent variable is 
always the whole of the asserted proposition in which it occurs, even when, 
typographically, its scope appears to be only part of the asserted proposition. 
Thus when fax) . <f>x or (x) . <f>x appears as part of an asserted proposition, it 
does not really occur, since the scope of the apparent variable really extends 



SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 131 

to the whole asserted proposition. It will be shown, however, that, so far as the 
theory of deduction is concerned, (gar) . $x and (a?) . <f)X behave like propositions 
not containing apparent variables. 

The definitions of implication, the logical product, and equivalence are to 
be transferred unchanged to (x) . <f>x and (g#) . <f>x. 

The above definitions can be repeated for successive types, and thus reach 
propositions of any type. 

Primitive Propositions. The primitive propositions required are six in 
number, and may be divided into three sets of two. We have first two 
propositions, which effect the passage from elementary to first-order proposi- 
tions, namely 

*91. b-.(f>x.D.(^z).<f>z Pp 

*911. b : <j>x v <f>y . D . (rz) . <f>z Pp 

Of these, the first states that, if <f>x is true, then there is a value of <f>z~ 
which is true; i.e. if we can find an instance of a function which is true, then 
the function is "sometimes true." (When we speak of a function as "some- 
times" true, we do not mean to assert that there is more than one argument 
for which it is true, but only that there is at least one.) Practically, the above 
primitive proposition gives the only method of proving "existence-theorems": 
in order to prove such theorems, it is necessary (and sufficient) to find some 
instance in which an object possesses the property in question. If we were to 
assume what maybe called "existence-axioms," i.e. axioms stating (qz) . <j>z for 
some particular $, these axioms would give other methods of proving existence. 
Instances of such axioms are the multiplicative axiom (#88) and the axiom of 
infinity (defined in #120-03). But we have not assumed any such axioms in 
the present work. 

The second of the above primitive propositions is only used once, in 
proving (rz) .<f>z.v. (rz) .<f>z:5. (qz) . <f>z, which is the analogue of #12 
(namely pvp.O .p) when p is replaced by faz). <f>z. The effect of this 
primitive proposition is to emphasize the ambiguity of the z required in order 
to secure (rz) . $z. We have, of course, in virtue of #9-1, 

<f>x . D . (a^) . <|z and <f>y . D . faz) . <f>z. 

But if we try to infer from these that <f>x v <£y . D . faz) . <f>z, we must use the 
proposition qDp .rOp .3 .qv rDp, where p is (gs) . <£z. Now it will be 
found, on referring to *4'77 and the propositions used in its proof, that this 
proposition depends upon #12, i.e. pvp.D .p. Hence it cannot be used by 
us to prove (ga?) . <f>x . v . (ga?) . ^x : D . (ga;) . <j>x, and thus we are compelled 
to assume the primitive proposition *9'11. 

We have next two propositions concerned with inference to or from propo- 
sitions containing apparent variables, as opposed to implication. First, we have, 

9—2 



132 MATHEMATICAL LOGIC [PART I 

for the new meaning of implication resulting from the above definitions of 
negation and disjunction, the analogue of *1'1, namely 
*9'12. What is implied by a true premiss is true. Pp. 

That is to say, given "h .p" and "\-.pDq" we may proceed to "h . q" 
even when the propositions p and q are not elementary. Also, as in ♦I'll, we 
may proceed from "H . <f>x" and " h . <f>x D ^x" to " h . tyx" where a; is a real 
variable, and <f> and ^ are not necessarily elementary functions. It is in this 
latter form that the axiom is usually needed. It is to be assumed for functions 
of several variables as well as for functions of one variable. 

We have next the primitive proposition which permits the passage from a 
real to an apparent variable, namely "when <f>y may be asserted, where y may 
be any possible argument, then (x) . <f>x may be asserted." In other words, when 
<f>y is true however y may be chosen among possible arguments, then (x) . <j>x 
is true, i.e. all values of <f> are true. That is to say, if we can assert a wholly 
ambiguous value $y, that must be because all values are true. We may express 
this primitive proposition by the words : " What is true in any case, however 
the case may be selected, is true in all cases." We cannot symbolise this pro- 
position, because if we put 

"\-:<f>y.3.(x).<l>x" 

that means: "However y may be chosen, <f>y implies (x) . <f>x," which is in 

general false. What we mean is: "If <f>y is true however y may be chosen, then 

(x) . <f>x is true." But we have not supplied a symbol for the mere hypothesis 

of what is asserted in " r . 4>y" where y is a real variable, and it is not worth 

while to supply such a symbol, because it would be very rarely required. If, 

for the moment, we use the symbol [<f>y] to express this hypothesis, then our 

primitive proposition is 

H : [<f>y] . D . (x) . (f>x Pp. 

In practice, this primitive proposition is only used for inference, not for impli- 
cation; that is to say, when we actually have an assertion containing a real 
variable, it enables us to turn this real variable into an apparent variable by 
placing it in brackets immediately after the assertion-sign, followed by enough 
dots to reach to the end of the assertion. This process will be called "turning, 
a real variable into an apparent variable." Thus we may assert our primitive 
proposition, for technical use, in the form: 

*913. In any assertion containing a real variable, this real variable may be 
turned into an apparent variable of which all possible values are asserted to 
satisfy the function in question. Pp. 

We have next two primitive propositions concerned with types. These 
require some preliminary explanations. 

Primitive Idea: Individual. We say that x is "individual" if a; is neither 
a proposition nor a function (cf. p. 51). 



SECTION B] EXTENSION OF THE THEORY OP DEDUCTION 133 

*9 131. Definition of "being of the same type." The following is a step-by-step 
definition, the definition for higher types presupposing that for lower types. 
We say that u and v "are of the same type" if (1) both are individuals, (2) both 
are elementary functions taking arguments of the same type, (3) u is a function 
and v is its negation, (4) u is <f>x or ^Sb, and v is $c v tylb, where <pfc and tyx 
are elementary functions, (5) u is (y) . <£ (&, y) and v is (z) . ty (x, z), where 
4* (P* P)> ^ (^» 9) are °f tne same type, (6) both are elementary propositions, 
(7) u is a proposition and v is ~w, or (8) u is (#) . <f>x and v is (y) . ^ry, where 
<£a\and ^& are of the same type. 

Our primitive propositions are: 
*9'14. If "<f>x" is significant, then if x is of the same type as a, "<j>a" is 
significant, and vice versa. Pp. (Cf. note on *10*121, p. 140.) 

*915. If, for some a, there is a proposition <f>a, then there is a function $&, 
and vice versa. Pp. 

It will be seen that, in virtue of the definitions, 

(x) . §x . "D . p means ~(a?) .<f>x.v.p, i.e. (gar) .~<f>x . v . p, 

i.e. (gyi?) .<^<f>xvp, i.e. (gai) . <$>x Op 

(ga?) *<f>x .0 .p means ~(a#) .<f>x.v.p, i.e. (x) .~<f>x . v . p, 

i.e. (x).<*j<f)xvp, i.e. (x) . <f>x D p 

In order to prove that (x) . $x and (ga;) . <fyx obey the same rules of deduction 
as (ftx, we have to prove that propositions of the forms (x) . §x and (ga;) . <f>x 
may replace one or more of the propositions p, q, r in #1'2 — *6. When this has 
been proved, the previous proofs of subsequent propositions in #2 — #5 become 
applicable. These proofs are given below. Certain other propositions, required 
in the proofs, are also proved. 

*9'2. h : (a?) - ^wzr - D - 

The above proposition states the principle of deduction from the general 
to the particular, i.e. "what holds in all cases, holds in any one case." 

Dem. 

h.*2-1.0h.~<£yv<f>y (1) 

H.*9*l ^Dhr«v>0yv^y.D.(aa?).~^»v^ (2) 

h.(l).(2).*l-ll.Dl-.(aa;).~^v^ (3) 

1 [(3).(*9-05)] r- : (a«) . ~ <f>x . v . <f>y (4) 
[(4).(*9-01.*r01)] \-:(x).<f>x.0.<l>y 

In the second line of the above proof, " <^> ^>y v <j>y" is taken as the value, 
for the argument y, of the function " ~ <f>x v <f>y," where x is the argument. 
A similar method of using #91 is employed in most of the following proofs. 

•'♦I'll is used, as in the third line of the above proof, in almost all steps 
except such as are mere applications of definitions. Hence it will not be 



134 MATHEMATIcili I#GIC [PART I 

further referred to, unless in oases where its employment is obscure or specially 
important. 

*9'21. I- :.(x) .<f>x"Dyfrx . D :(#)■ 4>x . ^ •(&)' V ra? 

J.e. if <f>x always implies yjrx, then "<f>x always" implies "yfrx always." The 
use of this proposition is constant throughout the remainder of this work. 

Dem. 

h.*2-08. Db:<f>zD^z,D.<f>z^^rz (1) 

h . (1) . *91 . Db:fay):<f>zDylrz.D.<f>yDyfrz (2) 

K(2).*9'l. Dh:.fax):.(^y):4>xDyjrx.D.<f>yDiJrz (3) 

h.(3).*913. Dh::(z)::fax):.fay)-.(l>x^yjrx.D.(f>yD^z (4) 

[(4).(*9'06)] \-::(z)::fax):.<f>xDylrx.D:fay).<f)yD^z (5) 

[(5).(*r01 .*9-08)] h : . fax) . ~ (<f>x D yjrx) : v : (z) : fay) .^^yvyfrz (6) 

[(6).(*9'08)] h :. fax) . ~ (<f>x D yjrx) : v : fay) . ~ <f>y . v . (s) . ^ (7) 

[(7).(*1'01)] Y:.(x).<\>x^^x.^'.(y).4>y.^.(z).y\rz 

This is the proposition to be proved, since "(y) . <f>y" is the same propo- 
sition as "(as) . <f)X," and "(z) . tyz" is the same proposition as "(x) . yjrx." 

*9'22. t-:.(x).<f)xDylrx. D:fax).<f>x. D.fax).tyx 

I.e. if <f>x always implies tyx, then if <f>x is sometimes true, so is tyx. This 
proposition, like #921, is constantly used in the sequel. 

Dem. 

K*2'08. D H : 0jO fy . D . <£y D ^ (1) 

K(l).*91. Db:faz):<f>yDfy.D.<f>yDfz ,(2) 

h.(2).*91. D\-:.fax):.faz):<f>xDfx.D.<l}y^fz (3) 

I- . (3) . *913 . D\-::(y)::fax)'..faz):<j>x^yfrx.D.<f)yDylrz (4) 

[(4).(*9'06)] h :: (y) :: fax) :.<f>xD+x.D : faz) .cfyyO^z (5) 

[(5).(*r01.*9-08)] \-::fax).~(<f>xDylrx):v:(y):faz).<l>yDylnz (6) 

[(6).(*101.*9'07)] I" " (3«) • ~(^ D ^-a?) : v : (y) . ~<£y . v . (gs> . ^* (7) 
[(7).(*101.^9-0V02)]b:.(x).<f>xDy{rx.D:fay).<f>y.O.faz).yJrz 

This is the proposition to be proved, because fay) . <j>y is the same pro- 
position as fax) . <f>x, and faz) . -tyz is the same proposition as fax) . yfrx. 

*923. H:(ar).^B.D.(*).^c [Id . *913-21] 
*9 24. h : fax) .<f>x.^. fax) . <f>x [Id . *9\L3-22] 
*925. \-:.(x).pv<f>x.D:p.v.(x).<j>x [*9"23 . (*9 04)] 

We* are now in a position to prove the analogues of #1*2 — "6, replacing 
one of the letters p, q, r in those propositions by (x) . <j>x or fax). <f>x. The 
proofs are given below. 



SECTION B] EXTENSION OP THE THEORY OF DEDUCTION 135 

#9'3. H :.(«?). <f>x. v.(x).<f>xi 3. (a;). 0a* 
Bern. 

h.*l'2. Dh.^v^.D.^ (1) 

K(l).#91. Dh:(gy):<£a*v0y.3.<£a; (2) 

K(2).*913. D>:.(a-):.(gy):<£a*v</>y.D.</>a- (3) 

[(3).(*9-05-01-04)] V :. (a*) : . <f>x . v . (y) . <£y : D . <j>x (4) 

h . (4) . #9*21 . y\-:.(x):<f>x.v.(y).<l>y:D.(x).<f)x (5) 

[(5).(*9-03)] h :. (a?) . <£a? . v.(y).<f>y: D ..(a?) . <£a- :. D H . Prop 

#9*31. h :. (gar) . <*Sa* . v . (ga*) . <£a* : D . (gas) . <f>x 

This is the only proposition which employs #9*11. 

Dem. 

h. #9*1113. Db:(y):<f>xv<l>y.D.(^).<f>z (1) 

[(l).(*9-03-02)] h : (gy) .<f>xv<l>y.D. (gs) . <f>z (2) 

h.(2).*9-13.DH:(a;):( a2 /).«/>a ; v^.D.(^).^ (3) 

[(3).(*90302)] r :. (ga*) : (gy) ■ ^ v <£y : D . (gs) . £* (4) 

[(4).(*905*06)] 1- :. (ga*) . <f>x . v . (gy) . </>y : D . (g*) . </>* 
#9*32. h:.^.D:(a?).^>a?.v.^ 

Dem. 

K*l*3. 3h:.q.D:<l>x.v.q (1) 

h . (1) . #913 . D h :. (x) :. ? . D : fs . v . q 
[*925] Dr:.g.D:(a*):<£a*.v.2 (2) 

[(2).(*903)] h:.q.O:(x).<f)X.y.q 

#9*33. r- :. g- . D : (ga*) ,<f>x.v.q [Proof as above] 
#9*34. h :. (a?) . <j&a- . D : /> . v . (x) . </>a* 

Dem. 

h.*l-3. 3\-:4>x.D.pv<f>x (1) 

1- . (1) . #913 . D\-i(x):(f>x.D.pv<f>x (2) 

K(2).*9*21. D\-:(x).(f>x.D.(x).pv<f>x (3) 

I- . (3) . (#904) . D r . Prop 

#9*35. h : . (ga;) . <f>x . D : p . v . (ga:) . <£a* [Proof as above] 

#9*36. \".. p . v . (x) . <j>x : : (x) . <f>x . v . p 

Dem. 

H.*l-4. D\- :pv <f>x."D .<j>xvp (1) 

H . (1) . #913*21 . D\-:(x).pv<f>x.D.(x).<l>xvp (2) 

I- . (2) . (#903*04) . D h . Prop 
#9*361. I- :. (x) . <j>x . v . p : D : p . v . (a;) . cf>x [Similar proof] 
*9'37. h :.^>. v.(ga?). <£a*: D :(ga*). <f>x . v.p [Similar proof] 
#9*371. r- :. (ga?) .<f>x.v ,p:D:p .v . (ga;) . <px [Similar proof] 



136 



MATHEMATICAL LOGIC 



[PART I 



*9'4. I- ::p : v : q . v . (x) . <f>x :. D :. q : v :p . v . (x) .<f>x 
Dem. 

r-.*l'5.*9'21 . Dh.(*):j).v.gv^s 3; (a;) :q . v .pv<f>x 
l-.(l).(*9-04).DH.Prop 
*9'401. I- :: p : v : q . v . fax) . <f>x :. >:. q:v:p.v . fax) . <£a; 
r- :: p : v : (x) . <f>x . v . r :. D :. (x) . <$>x : v : p v r 
H :: p : v : fax) . <£a? . v . r :. D :. fax) . <£# : v : p v r 
I- :: (x) . <}>x : v : qv r z. D :. q : v : (x) . <f>x . v . r 
\- :: fax) . §x : v : q v r :. D :. g : v : (ga?) . <j>x . v . r 
h ::p D q . D :. p . v . (x) . (fix : D : q . v . (x) . <$>x 



(1) 



*941. 

*9'411 

*942. 

*9421 

*95. 

Dem. 
r . #1-6 . D h 

H.(l).*9-l.(*9-06). Dh 
K(2).*913.(*904).Dr 
[(3).(*9-08)] r 

[(4).(*9'01)] J- 

[(5).(*904)] I- 



[As above] 
[As above] 
[As above] 
[As above] 
[As above] 



:.pDq.D:py<j>y.D.qv<f>y (1) 

:.pD q. D :fax)zpv<f>x. D .qv<f>y (2) 

::p D 9 . D :. (y) :. (ga;) :p v <£#, 0.qv<j>y (3) 
::p D g . D :. (g#).~(p v <fcc) . v . (y) . q v <f>y <4) 
r.pDq. D :.(x). pv <f>x . D. (y)-qv <]>y (5) 

:: p 3 <? . 3 :. p • v . («) . <f>x : D : q . v . (i/) . <py 

[As above] 



*9501. h ::p >g . D :.p . v . (ga?) . <£a; : D : g . v . (ga;) . $a; 
$9*51. f- ::p . D . (x). <j)x: "Dt.pvr. 3 : (x).<f>x. v.r 
Dem. 

K#r6. D I- :.pD"<j>x. D :pvr. D. ^rvr (1) 

h . (1) . *913-21 . Db:'.(x).pD<f>x.3:.(x):pvr.D.tf>xvr (2) 
H . (2) . (*90304) . D h . Prop 
*9'511. h :: p . D . fax) . <£# : D :. p v r . D : (ga?) . <f>x . v . r [As above] 
#952. h ::'(«) . <f>x . D . q : "D :. (x) . <}>x .V . r : 2 . qv r ~ 
Dem. 

K#r6. D H:.^a?D^.D :<£a?vr . D.gvr (1) 

H . (1) . *91322 . 3b::fax).4>xDq.O:.fax)z<l>xvr.D.qvr (2) 
K (2) . (*9-05-01) . D H :: (a;) . <j>x . D . g : D :. («).^vr.D . q vr (3) 
I- . (3) . (*903) . DH.Prop 
*9 521. \- :: fax) . <f>x . D . q : D :. (ga;) . ^>ar . v . r : D . g v r [As above] 
*9"6. (a;) . <j>x, ~(a?) . ^>a;, fax) . $x and ~(ga;) . <j>x are of the same tjpe. 

[*9*131,(7)and(8)] 
*9"61. If $e and tyic are elementary functions of the same type, there is a 
function ^vi|r^. 
Dem. 

By *9'14*15, there is an a for which "^ra," and therefore "<f>a," are 
significant, and therefore so is "<j>a v tya," by the primitive idea of disjunction. 
Hence the result by *915. 

The same proof holds for functions of any number of variables. 



SECTION B] EXTENSION OF THE THEOBY OF DEDUCTION 137 

*9 62. If <£(£, §) and <fz are elementary functions, and the ^-argument to 
^> is of the same type as the argument to yjr, there are functions 

iy) .<}>($,y).v. ^a, (a^) . <t> (^» y) - v • ^> 

Dem. 

By #9*15, there are propositions <f>(x, b) and tya, where by hypothesis % 
and a are of the same type. Hence by #9*14 there is a proposition <f> (a, b), 
and therefore, by the primitive idea of disjunction, there is a proposition 
<[>(a,b)v >jra, and therefore, by *9'lo and #9'03, there is a proposition 
(y) . <f> (a, y) . v . yfra. Similarly there is a proposition (;jy) . <f> (a, y) . v . ifra. 
Hence the result, by *9*15. 

*9'63. If $ (&, $), i/r (&, P) are elementary functions of the same type, there 
are functions (y) . <f> ($, y) . v . (*) . yfr (&, z), etc. [Proof as above] 

We have now completed the proof that, in the primitive propositions of 
*1, any one of the propositions that occur may be replaced by (x) . <j>x or 
(a#) . <f>x. It follows that, by merely repeating the proofs, we can show that 
any other of the propositions that occur in these propositions can be simul- 
taneously replaced by (#) .-tyx or (g#) . ifrx. Thus all the primitive propositions 
of *1, and therefore all the propositions of *2 — *5, hold equally when some 
or &\l of the propositions concerned are of one of the forms (%) . <f>oc, "(ga?) . <£#, 
which was to be proved. 

It follows, by mere repetition of the proofs, that the propositions of #1 — *5 
hold when p, q, r ape replaced by propositions containing any number of 
apparent variables. 



*10. THEORY OF PROPOSITIONS CONTAINING 
ONE APPARENT VARIABLE 

Summary of *10. 

The chief purpose of the propositions of this number is to extend to 
formal implications (i.e. to propositions of the form (x) . §x D yjrx) as many as 
possible of the propositions proved previously for material implications, i.e. 
for propositions of the form pDq. Thus e.g. we have proved in *3*33 that 

pOq.qDr.D.p'Dr. 
Put p = Socrates is a Greek, 

q = Socrates is a man, 
r = Socrates is a mortal. 

Then we have " if ' Socrates is a Greek ' implies ' Socrates is a man,' and 
' Socrates is a man ' implies ' Socrates is a mortal,' it follows that ' Socrates is 
a Greek ' implies ' Socrates is a mortal.' " But this does not of itself prove 
that if all Greeks are men, and all men are mortals, then all Greeks are 
mortals. 

Putting <fcx . = . x is a Greek, 

sjrx . — . x is a man, 

XX . = . x is a mortal, 
we have to prove 

(x) . <$>x D yfrx : (x) . yfrx D %x : D : (x) . <f>x D %x. 

It is such propositions that have to be proved in the present number. It will 
be seen that formal implication ((x) . <f>x D ^x) is a relation of two functions 
0& and -tytb. Many of the formal properties of this relation are analogous to 
properties of the relation "p D q " which expresses material implication ; it is 
such analogues that are to be proved in this number. 

We shall assume in this number, what has been proved in #9, that the 
propositions of *1 — *5 can be applied to such propositions as (x) . <px and 
(gaj) . <f>x. Instead of the method adopted in *9, it is possible to take negation 
and disjunction as new primitive ideas, as applied to propositions containing 
apparent variables, and to assume that, with the new meanings of negation 
and disjunction, the primitive propositions of #1 still hold. If this method is 
adopted, we need not take fax) . <f>x as a primitive idea, but may put 
*1001. ( l 3/c).<f)x. = .'^(x).^>^x Df 

In order to make it clear how this alternative method can be developed, 
we shall, in the present number, assume nothing of what has been proved in 
*9 except certain propositions which, in the alternative method, will be 
primitive propositions, and (what in part characterizes the alternative method) 



SECTION B] THEOBY OP ONE APPABENT VARIABLE 139 

the applicability to propositions containing apparent variables of analogues 
of the primitive ideas and propositions of #1, and therefore of their conse- 
quences as set forth in #2— #5. 

The two following definitions merely serve to introduce a notation which 
is often more convenient than the notation (x) . $x D tyx or (x).<px = yfrx. 
*1002. <f>xD x -^x. = .(x).<f>xDylrx Df 
*10 03. <f>x = x tyx . — . (x) . <f>x = yfrx. Df 

The first of these notations is due to Peano, who/however, has no notation 
for (x) . $x except in the special case of a formal implication. 

The following propositions (*10ri 1/1 2'1 21 1 22) have already been given 
in *9. *101 is *9'2, *1011 is *913, *1012 is *925, *10121 is *914, and 
*10*122 is *9"15. These five propositions must all be taken as primitive 
propositions in the alternative method; on the other hand, #91 and #9 , 11 are 
not required as primitive propositions in the alternative method. 

The propositions of the present number are very much used throughout 
the rest of the work. The propositions most used are the following: 
*101. h : (a;) . <£# . D . <£y 

I.e. what is true in all cases is true in any one case. 
¥1011. If <f>y is true whatever possible argument y may be, then (x) . <f>% is 
true. In other words, whenever the propositional function <py can be asserted, 
so can the proposition (x) . <f>x. 

*10'21. h :. (x) . p D <f>x . = : p . D . {as) . <f>x 
*10'22. h :. (x) . <f>x . yfrx . = :(x). <f>x: (x).tyx 

The conditions of significance in this proposition demand that <j> and yfr 
should take arguments of the same type. 

*1023. b:.(x).<f>x'2p. = :('2Lx).<j)x.D.p 

I.e. if <f>x always implies p, then if <f>x is ever true, p is true. 

*10-24. h : <j>y . D . (roc) . <f>x 

I.e. if $y is true, then there is an x for which <f>x is true. This is the sole 

method of proving existence-theorems. 

*10 27. H :. (z) . <f>z D tyz . D : (z) . <f>z . D . (z) . tyz 

I.e. if <j>z always implies -tyz, then " <f>z always " implies " yjrz always." The 

three following propositions, which are equally useful, are analogous to *10"27. 

*10-271. f- :. (z) . 4>z ~ fz . D :(z) .<f>z . = .{z).tyz 
*10'28. V :.(x). <f>x'2i{rx. D : fax). <f>x . D . fax).yjrx 
*10'281. H :. (x) . (f>x = yfrx . D : fax) ,<f>x. = . fax) . ijrx 
#1035. h :. fax). p. <j>x. = :p : fax) . <f>x 
*10'42. h :. fax) .<f>x.v. fax) . -fx : = . fax) .<f>xvyfrx 
#10 - 5. h : . fax) . <j>x . fa . D : fax) . <f>x : fax) . yfrx 



140 MATHEMATICAL LOGIC [PART I 

It should be noticed that whereas #10'42 expresses an equivalence, #10*5 
only expresses an implication. This is the source of many subsequent 
differences between formulae concerning addition and formulae concerning 
multiplication. 

#10*51. h :.. ~ {(gas) • <f>x • tyx) • = : </># . '3* . ~ tyx 
This proposition is analogous to 

h s ~ (p . q) i = . p 3 <>*q 
which results from *4*63 by transposition. 

Of the remaining propositions of this number, some are employed fairly 
often, while others are lemmas which are used only once or twice, sometimes 
at a much later stage. 

#1001. (ga*).<j!>#. = .~(a i ).~<£a* Df 

This definition is only to be used when we discard the method of *9 in 
favour of the alternative method already explained. In either case we have 

h : (ga;) .<f>x. =■ . ~(a*) . ™$tx. 
#1002. ^>xD x yjrx,.= .{x) m ^D^rx Df 
#10*03. <j>x= x '\Jrx. — .(x).<f>x = $'X Df 
#10*1. h : (x) . <f>x .3 . $y [*9*2] 

#10*11. If 4>y is true whatever possible argument y may be, then (x).<f>x is 
true. ^*9-l3] 

This proposition is, in a sense, the converse of #10*1. #101 may be stated : 
" What is true of all is .true of any," while #10:11 may be stated : " What is 
true of any, however chosen, is true of all." 

#1012. h : . \x) . p v <f>x, 3.: p . v . (a?) . tf>x [#9*25] 

According to the definitions in *9, this proposition is a mere example 
of "qSq," since tby definition the two sides of the implication are different 
symbols for the same proposition. According to the alternative method, on 
the contrary, #10*12 is a substantial proposition. 

#10*121. If " ifxc" is significant, then if a is of the same type as x, '*#«" is 
significant, and vice versa. •■[#D:1 4] 

It follows from this proposition that two arguments to the same function 
must be df the same type; for if a? and a are arguments to $£, "(j>x" and "<f>d" 
are significant, ^and therefore a; "and a are of the same type. Thus. the above 
primitive proposition embodies the outcome of our discussion of the vicious- 
circle paradoxes in Chapter II of the Introduction. 

#10122. If, for some a, there is a proposition <fm,then there is a function ^c, 
And vice versa. [*9'I5J 

#1013. If $& and ^ take arguments.of the-same type, arid we have "' kJ$x" 
and " H .tyx? we shall have *■ kvxf>x . yfrx." 



SECTION b] theory of one apparent variable 141 

Bern,. 

By repeated use of *96r62-63131 (3), there is a function ~^v~^. 
Hence by *211 and *301, 

h : <^<f>xv<^y^x. v . <f>x.tyx (1) 

h.(l).*2-32.(*l-01).Dh:.^.D:'«/ra;.3.^,^ (2) 

h . (2) . *912 . D K Prop 

*1014 h :.(#).<£#: (a?). ^arO .<f>y.yjry 

This proposition is true whenever it is significant, but it' is not always 
significant when its hypothesis is significant. For the thesis demands that 
<f> and -\fr should take arguments of the same type, while the hypothesis does 
not demand this. Hence, if it is to be applied when <£ and -^ are given, or 
when yfr is given as a function of or vice versa, we must not argue from the 
hypothesis to the thesis unless, in the supposed case, # and ty take arguments 
of the same type. 

Dem. 
K*10-l. D\-z(x.).<f>x.y.<l>y (I) 

K*10\L. Dhr^.^.D.-ifry (2) 

H . (1) . (2) . *1013 . D h : (x) . <jxc . D . fyy z (x)'. -far . D . yfr y z 
[*3*47] ^ D\-i.{x).<f)x:(x).'fx:'2>.<f}y.'^y m .. DKProp 

*102. \-:.(x).pv $x.= :p.v .(x).<l>x 

Dem. 

h . *10'1 . *l-6 . D H :. p . v . (x) . <f>x : D . p v <f>y :. 
[*1011] D f :-;.(y) :. p . v . (a?) . 0a? :0.pv<f>y z. 

[*1012] yh:.p.v.(x).<j>x:D.(y).pv<l>y (1) 

K*1012. 3>:.(y)._pv^y.3:p.v.(a?).^aj (2) 

K(l),(2). Dr.. Prop 

*1021. h :..(#)■ 2> ^ </>*• = =2>- 3- (*0-<^ [*10-2^1 

This proposition is much more used than #10'2. 

*10'22. H :. (x) . <f>x . yjrx . = :(#).<£#:(#)■ yjcx 
Dem. 

h . *10'1 . D H : (x) . $x . ^rx . D . ^y . yfry . (1) 

[*3-26] D.^y: 

[*1011] DI-:.(y):(ar).^.^.D.^:. 

[*10-21] >.h:.(#)/^.^.D.(y).<£y (2). 

h.(l).*3'27. y\-i.\x).<\>x.^x.^.^zi. 

[*1011] D I- : . (z) z (x) . $x . yjrx . D . -fz z . 

[*10'21] D\-:.(x).sf>x.ylrx.D.(z).fz (3) 

t- . (2) . (3) . Comp .y\-:.(x).(f>x.ylrx.D: (y) . <f>y z (z) . ^ (4) 

h . *1014-11 . Dhz. (y) :. (#) .<f>xz(x) .yjtxzD .<f>y .yfryz. 

[*1Q-21] D\-:.(x).^xz(x).^x:D.(y).4>y.iry (5) 

K(4).(5). Dh.Prop 



142 MATHEMATICAL LO0JC [PART I 

The above proposition is true whenever it is significant; but, as was 
pointed out in connexion with #1014, it is not always significant when 
" (#).<£#: (x) . yjrx " is significant. 

#10*221. If <fxc contains a constituent %(#, y, z, ...) and yjrx contains a con- 
stituent x ( x > u > v> ...), where x ia an elementary function and y, z, ... w, t>, ... 
are either constants or apparent variables, then <f>£ and tyS; take arguments 
of the same type. This can be proved in each particular case, though not 
generally, provided that, in obtaining <f> and yfr from x> X * s on ty submitted 
to negations, disjunctions and generalizations. The process may be illustrated 
by an example. Suppose <f>x is (y) . x (x, y ) . D . 0x, and yfrx is fx . D . (y ) . x (®> y)- 
By the definitions of #9, <j>se is fay) • ~% (x, y) v 0x, and yfrx is (y) . ~y<r v ^ (#, y)> 
Hence since the primitive ideas (x) . Fx and fax) . Fx only apply to functions, 
there are functions ~#(&, p)v0$, ~/^v^(^, $). Hence there is a proposi- 
tion ~%(a, b)v 0a. Hence, since "pvq" and "f^p" are only significant 
when p and q are propositions, there is a proposition x ( a > &)• Similarly, for 
some u and v, there are propositions ~/w v x (u, v) and x ( u > v )- Hence by 
#9*14, ii and a, v and b are respectively of the same type, and (again by #9*14) 
there is a proposition ~/jv^(a, b). Hence (#9'15) there are functions 
~X ( a > 9) v & a > ~/ a v X ( a > P)> a11 ^ therefore there are propositions 

(wf) • ~x ( a > y) v 0a > (y) ■ ~fa v x ( a > y)> 

i.e. there are propositions <f>a, yfra, which was to be proved. This process can 
be applied similarly in any other instance. 

#10*23. b:.(x).<f>x^p. = :fax).<f>x.D.p 
Dem. 

h . #4-2 . (#9'03) . D h :. (x) . ~<£a; v p . = : (x) . ~<f>x . v . p : 

[(*902)] s.(a*).^.D-i> (1) 

h . (1) . (#1-01) . DKProp 

In the above proof, we employ the definitions of #9. In the alternative 
method, in which fax) . <f>x is defined in accordance with #10-01, the proof 
proceeds as follows. 

#10*23. \-:.(x).(f>xDp. = : fax) . <f>x . D. . p 
Dem. 
b . Transp . (#10-01) . D I- :. fax) . <f>x . D . p : = : ~p . D . (x) . ~<f>x : 
[#10-21] =:(a;):~jj.D.~^r: (1) 

[#101] D:~p.D.~<j)x: 

[Transp] D :<f>x .D .p :. 

[#10-11] D I- :. (x) :. fax) .<f>x. D . p .O : <f>x . D . p :. 



SECTION B] THEORY OF ONE APPARENT VARIABLE 143 

[*10'21] DH:.(g^).^c.D.j?0:(a;):f».D.p (2) 

b . #10'1 . Dh:.(jr):^.O.p:D:^ Dp : 
[Transp] D:~j3.D.^^a;i. 

[*10'11'21] 3b:.(x):<f>x.D.p:'2:(x)z~p. , 2.~<f>x: 
[(!)] D:(a^).^.D.p (3) 

K(2).(3). DKProp 

Whenever we have an asserted proposition of the form p D <f>x, we can 
pass by *10*11'21 to an asserted proposition p . D . (x) . <£wc. This passage is 
constantly required, as in the last line but one of the above proof. It will 
be indicated merely by the reference " #10*1 1 '21/' and the two steps which it 
requires will not be separately put down. 

♦1024. f- : <j>y . D . fax) . <f>x 

This is *9'1. In the alternative method, the proof is as follows. 

Dem. 

K*101 .Oh:(a;).~^.D.~^: 

[Transp] D h : <f>y . D . ~(#) . ~<f>x : 

[(*10-01)] D I- . Prop 
♦1025. b:(x).<f>x.D.fax).<j>x [*10-124] 

*10 251. b : (x) . ~ <f>x . D . ~ {(x) . <£#} [*1025 . Transp] 
♦10252. h :~{fax) . <£#} . = . (x).~<f>x [*4«2 . (*9'02)] 
♦10 253. h : ~ \{x) . <f>x] . = . fax) .~<j>x [*4"2 . (*901)] 

In the alternative method, in which fax). <f>x is defined as in fclO'01, the 
proofs of *10'252-253 are as follows. 

*10-252. b : ~ [fax) . <f>x} . == . (x) . ~ <f>x [*413 . (*1001)] 
♦10 253. b : ~ {(x) .<f>x}. = . fax) . ~ <f>x 



Dem. 



K*10'l. Dh:(a?).^c,D,^y. 

[*2-12] D.~(~0y): 

[♦101121] D h : (x) . <f>x . D . (3/) .~(~tf>*/) : 

[Transp] D h : ~ {(y) . ~ ( <-o ^>y)} . D . ~ {(#) . <£#} : 

[(♦10-01)] D I- : (ay) .~*y . ■ D .~{(*) . <M (1) 

K*101. DH:(y).~(~03/). D.~(~<£#)» 

[♦2-14] 2.<j>x: 

[*1011-21] 3 h : (y) -~(~#) . D . (x) . <f>x : 

[Transp] D b : ~ {(#) . <f>x} . D . ~ {(3/) . ~ ( ~ $3/)} . 

[(♦1001)] D.(ay).~*y (2) 

K(l).(2).Dh.Prop 



144 MATHEMATICAL LOGIC [PART I 

*1026. b:.(z).<f>zyyjrz:<f)xiy.fa; [*101 . Imp] 

This is one form of the syllogism in Barbara. E.g. put <f>z . = . z is a man, 
■tyz . = . z is mortal, x = Socrates. Then the proposition becomes: 

"If all men are mortal, and Socrates is a man, then Socrates is mortal." 

Another form of the syllogism in Barbara is given in *10*3. The two 
forms, formerly wrongly identified, were first distinguished by Peano and 
Frege. 
*10-27. h:.( < z).<f>zyylrz.D:(z).<f>z.D.(z).yfrz 

This is *9*21. In the alternative method, the proof is as follows. 

Dem. 

b . #10-14 . D h :. (z) . <f>z D yjrz : (z) * <f>z : D . 4>y D ijry • <f>y • 

[Ass] D . tyy :. 

[#10-1] D h :. (y) :. (z) . cf>z D yjrz : (z) . <j>z : D . yjry :. 

[*10'21] D h :. {z) . 0* D <fz : (*) . 0* : D . (y) . ^2/ C 1 ) 

I- . (1) . Exp . y r . Prop 

*10-271. h :. 0) . 0z = ^ . D :.(*) . 0* . = . (z) . ^z 

Dem. 

K*10'22. > h:.Hp. :>:(*). 0*3-0^ : 

[*1027] >:(z).<f>z.D.(z).yfrz (1) 

K*1022. ' DJ-:.Hp.D:(*).^eO0s: 

[*10'27] D:(s).^s.>.(.!r).0* (2) 

H . (1) . (2) . Comp . y h . Prop 

#10*28. h : . (x) . 0a; !> -0-a; . D : (ga;) . 0a; . D . (ga?) . ^a; 

This is #9-22. In the alternative method, the proof is as follows. 

Dem. 

h . *ioi . y\- : . (x) . <f)xy^-x .y.<f>y y^-y • 

[Transp] D . ~ tyy D ~ <f>y : . 

[*10\L1~21]3> :. {x).<f>x y tyx . y : (y) .~-fyy~<f>y : 
[*10-27] y:(y).~ylry.y.(y).~<f>y: 

[Transp] D : (ay) . 0y • 3 . (ay) , ^y « 3 r- . Prop 

*10 28L h : . (a;) .<j>x= tyx . O : (rx) . 0a; . = . (a«) . ^ra? [*10-22"28 . Comp] 
*1029. h :. (x) .<f>xy-fx :(x) . <f>x D %x : = : (x) : <f>x . D . yfrx . x x - 

Dem. 

h . *lQ-22 . D h :. (x) . 0a? D tyx : (a;) . 0a; D %a; : 

= : (ar) : 0a; D -^a; . 0# D ^a; (1) 

t- . #4*76 . D"r- :. 0a; D^ar* 0a; D ;•£*?. = : 0a; . D . ty x •X XZm 

[#1011] D K :. (a;) :. 0a; 3'i^tt'. 0a; D %a? . = : 0a; . D . -tyx . ^a; :. 

[#10'27l] y b :. (a?) : 0a;O ^ar. 0a; D %a? : = : (x) : 0a; . D . -^ra? . x x (2) 

r- . (1) . (2) .>+ . Prop 
This is an extension of the principle of composition. 



SECTION B] THEORY OF ONE APPARENT VARIABLE 145 

#10*3. h :. (x) . <f>x D yfrx : (x) . ^ra; D ^« : D . (#) . <£a" D x x 
This is the second form of the syllogism in Barbara. 
Dem. 

\- . #10*22*221 . D r : Hp . D . (x). <f>x D yfrx . yfrx D x x • 
[Syll .#1027] D . (x) .<f>xO X x;Oh. Prop 

#10*301. H :. (x) . <f>x = i/ra; : (a*) . i/ra? = %x : D . (a;) . <f>x = ^« 

I- . #1 0*22*221 . D I- :. Hp . D : (x) . <f>x = yfrx . yfrx = x x : 
[*4*22.*10*27] D : (a*) . 0a; = X a? :. D h . Prop 

In the second line of the proofs of #10*3 and #10*301, we abbreviate the 
process of proof in a way which is often convenient. In #10*3, the full process 
would be as follows: 

h . Syll . D r- : <f>x"D yfrx . yfrx D x x . D . <f>x D x x : 
[#10*11] D I- : (x) : <f>x D yfrx . yfrx D x x • ^ • $ x -^ X-^ : 
[#10*27] Dh(«).^Dfx,fa;D^.D. (a?) . <£# D %# 
The above two propositions show that formal implication and formal 
equivalence are transitive relations between functions. 

#10*31. V :. (x) . <f>x D yfrx . D : (x) : <f>x . x x . D . ^ra; . x x 
Dem. 

K Fact. #10* 11 . Dh :.(x):. <f>x^yfrx. D:<f>x.xx. ^.yfrx.xx (1) 
K(l). #10*27. Dr. Prop 
#10*311. I- :. (x) . <f>x = yfrx . D : (x) : <f>x . x x • = • ty x • X x 
Dem. 

r . #4*36 . #10*11 . D I- :. (x) :. $x = yfrx . 3 : <£a? . x x • = ■ "f® . xx (1) 
K(l). #10*27. Dr. Prop 
The above two propositions are extensions of the principle of the factor. 
#10*32. h : <f>x = x yfrx . = . yfrx = x §x 
Dem. 

V . #10*22 . D I- : <f>x = x yfrx . = . <f>x D x yfrx . yfrx D* <f)x . 
[*43] = . yfrx D x <f>X . <f>X Da, yfrx . 
[#10*22] = . yfrx =, ^v : D h . Prop 

This proposition shows that formal equivalence is symmetrical. 
#10*321. H : <f>x = x yfrx .<f>x= x x x '^- ty x -%X X 
Dem. 

V . #10*32 . Fact . D h : Hp . D . yfrx = x <f>x . <f>x = x xx . 
[#10301] D . yfrx = x X x : D h . Prop 

#10*322. I- : i/r# = x <f>a: . xx = x <f>x . D . \frx == x x x 
Dem. 

V . #1032 . D h : Hp . D . yfrx = x <j>x . <f>x = x xx . 
[#10*301 ] D . yfrx = x X x : D h . Prop 

R&W I 10 



14$ 



MATHEMATICAL LOGIC 



[PART I 



*10 33. I- :. (x) : <f>x .p : = : (x) .<l>x:p 

Dem. 

h.*10'l. Dh:.(x):<f>x.p:D.<f>y.p. 

[*3-27] D.p 

I- . (1) . *3'26 . D h :. (#):<£#. p :D . <£y : 

[*10-11-21] D I- :.(#): <f>x . p : D . (y) . <f>y 



(1) 
(2) 

(3) 

(4) 



(5) 



In the alternative 



K(2).(3). D\-:.(x):<f>x.piD:(y).<l>y:p 

K*101. D\-:.(y).<f>y. D.<jk»:. 

[Fact] Dh:.(y).^:p:D.fB.p:. 

[♦1011-21] 0\-'..(y).<f>y:p:0:(x):<f>x.p 

I- . (4) . (5) . D h . Prop 
*10-34. I- :. (g«) . <£aOp . = : (<c) . <j>x. D .p 

This follows immediately from *90501 and *1*01. 
method, the proof is as follows. 

Dem. 

K*4-2.(*10-01).D 

H:.( a #).<£aOjp. = :~{(x) .~(<f>xDp)} : 

[*461.*10'271] =:~{(a?):^r.~p|: 

[*1033] = : ~ {(«) • <j># : ~/>} : 

[*4-53] = : ~ {(*) ■ ty] ■ v .p '. 

[*46] s :(x).<f>x.D.p 

*10 35. H : . (ga;) .p.<f>x. = :p: (a#) .<£# 

H.*326. Dh:p.</>a:.D..p: 

[*1011] Dh:(#):p.<^.O.p: 

[*1023] D h : (a*) . p . <£a? . D . p (1) 

r- . *3-27 . Db-.p.Qx.D.fa: 

[*1011] D\-:(x):p.4>x.D.<f>x: 

[*10'28] Dh:(ga;).p.^c.D.(aa!).^ (2) 

K*3'2. "D\-:.p.Di<f>x.D.p.<f>x. 

[*1011-21] D t- :.p . D : (x) : <£#. D .p . <£# : 

[*10'28] D:(ga;).<^.3.(aa;).p.^B (3) 

|-.(1).(2).(3). Imp. Dh. Prop 

*1036. h:.(a«).«/>«vp. = :(a«).^.v.p 

This follows immediately from #905. In the alternative method, the 
proof is as follows. 



Dem. 

h . *4-64 . 

[*1011] 

[*10'281] 

[*1034] 

[*4'6.(*10-01)] 



D I- : <j>xvp . = ,~<j>x'Dp: 
D I- : (x) : <j>x vp . = . ~ <f>x Dp : 
D 1- :. (a«) .^vj). = : (ga?) .~<£# Dp : 
= : (x) ,f^><j)X. D .p : 
= : (3^) .f».v.])!.3 : K Prop 



SECTION B] THEORY OF ONE APPARENT VARIABLE 147 

The above proposition is only required in order to lead to the following: 



*10 37. I- :. (gar) . p 3 <f>x . = : p . 3 . (gar) . <j>a 



*1036 



P J 



#1039. h :. <f>x D x -x-x : ^ ^* 8x ' ^ : <fc» ■ ty x • ^x • X x • ® x 
Dem. 

r- . *10-22 . 3 f :. Hp . 3 : (a?) : <f>x 3 % « . -^ 3 0# : 
[*3-47.*10-27] D : (a) : cf>x . fco .0 .x x • Occ :.D b . Prop 

This proposition is only true when the conclusion is significant; the 
significance of the hypothesis does not insure that of the conclusion. On the 
conditions of significance, see the remarks on *10'4, below. 

#104. h :. <f>x = x xx .tyx = v 0x .D : <f>x . yfrx . = x . x x • @ x 

Dem. 
h . *10-22 . 3 1- : . Hp . 3 : <f>x 3* x x • "f * 3* Gx ' 

[*10-39] D:<f)X.yfrx.D x . x x -O x (1) 

Similarly h :. Hp . 3 : x x • & x • ^x- <f> x • ty x (2) 

I- . (1) . (2) . Comp . 3 h :, Hp . 3 : <f>x . tyx . D x . x x • @ x s X x • ® x • ^* • fa • "¥ x '• 
[*10-22] 3 : <\>x . yfrx . = x . x x ■ Bat :. 3 H . Prop 

In #10*4 and many later propositions, as in *10"39, the conclusion may be not 
significant when the hypothesis is true. Hence, in order that it may be legiti- 
mate to use #10*4 in inference, i.e. to pass from the assertion of the hypothesis 
to the assertion of the conclusion, the functions <f>, yjr, x* must be such as to 
have overlapping ranges of significance. In virtue of *1'0*221, this is secured if 
they are of the forms F{x, x (x,p,%...)}J{x,xM,%...)},0{x, X (x,'PX---)}, 
9 \ x > X ( x > §> $> •••)}• It is also secured if ^ and yfr or <f> and or x an( ^ ^ 
or x ar) d are of such forms, for $ and x must have overlapping ranges of 
significance if the hypothesis is to be significant, and so must -^ and 0. 

*10"41. h :.{x).$x. v .(x).yjrx: 3 . (#) . favyfrx 

Dem. 

h . *101 . 3 I- :(x).<f>x. 3 . <f>y . 

[*2-2] ^.cj>yvfy (1) 

r- . *101 . Dh:(*).^.D.^. 

[*l-3] D.^yv^y (2) 
h . (1) . (2) . *10 13 . 3 I- :. (a?) . <£# . 3 . <£y v yfry : (a-) . i/r# . 3 . ^ v yjry :. 

[#3'44] 3 h :. (.*) . $a . v . (x) . ^x : 3 . <f>y v ^y 

[*101121] 3 1- :. (a?) . <f>x . v . (a?) . yjrx : 3 . (y) . <jty v ^y :. 3 r- . Prop 

Observe that in the above proof the uses of #2*2 and *1*3 are only legitimate 
if <f>y and ^ry have overlapping ranges of significance, for otherwise, if y is such 
that there is a proposition <f>y, it is such that there is no proposition yfry, and 
conversely. 

10-.2 



148 MATHEMATICAL LOGIC [PART I 

*10"411. h :. <f>x = x j(x . yfros = x 0x . "D : <f>x v yjrx . = z . x® v Ox 
Dem. 

h . *10*14 . D h :. Hp . D : <£# = %x . yfrx = 0x : 

[*4'39] ^:<t>xvtyx. = .xxvdx (1) 

h.(l).*10-ir21.DKP.rop 

*10412. H- : ^c=, -fa; . = .r^^x= x <^^x [*4-ll . *1011271] 

#10-413. H :. <£# = x x x • ^ -« ^ ■ ^ : ^ 3 ^ x • =x • X 00 ^ ^ x 
Dem. 

r . *10-411-412 . 3 I- .-. Hp . D : ~$x v yjrx . = x . ~%# v 0x 
[(*1-01)] D:<f>xDylrx.= x . x xD6x:.Dh.T?rov 

*10414. h :. <f>x = x %x . tyx = x 0x . D : <f>x = i/r# . =3 . ^ = 6x 

Dem. 
H .*10413^|^|.*10-32 .Dr :.Hp .D : i/raO <£# .=.. $xD X x CO 

r . *10-413 . (1) . *10-4 . DKProp 

The propositions fclO'413414 are chiefly used in cases where either x i s 
replaced by <£ or is replaced by yjr, in which case half the hypothesis becomes 
superfluous, being true by #4*2. 

*1042. H :. fax) . <f>x . v . fax) . yjrx : = . fax) .Qxvfx 

Dem. 
h .*10"22 . D h :.(x).<»j<j)xz (x) .o^yfrx: = .(x) .~(])X .<^>yfrx :. 

[*4'11] D I- :.<>*>{(#) .<^><f>x: (x) .^yfrx] . = .~{(#) .<~<£#.~^r#} :. 

[*4'51'56.*10-271] D h :.~{(«) .~<^} . v .~{(» ,~^;} : 

= . ~ {(x) . ~ (<f>x v i|ra?)} :. 
[*10 # 253] D h :. (a#) . <px . v . (ga?) . i|ra? : = . fax) . <£# v yjrx :. 

D I- . Prop 

This proposition is very frequently used. It should be contrasted with 
#10*5, in which we have only an implication, not an equivalence. 

*10"43. h : <f)Z = z <fyz ,(f>x. = .<f>z= z yjrz . yfrx 

Dem. 

K*10\L. ^h:<f)Z= z yjrz.D.<f,x=yfrx (1) 

h . (1) . *5-32 . D H . Prop 

*105. b :. fax) . <f>x . yfrx . D:fax).<f>x: fax).\jrx 

Dem. 

b . *326 . #10-11 . D h : (x) : <j>x . -fx . D ,<f>x : 

[*10'28] Db:fax).(f>x.ylrx.D.fax).(l>x (1) 

h . *3'27 . *10'11 . D h :. (x) : <f>x . yjrx . D . tyx: 

[*10*28] Db:fax).<f>x.fx.D.fax).yJrx (2) 

h.(l).(2).Comp.Dl-:.Prop 



SECTION B] 



THEORY OP ONE APPARENT VARIABLE 



149 



The converse of the above proposition is false. The fact that this 
proposition states an implication, while #10*42 states an equivalence, is the 
source of many subsequent differences between formulae concerning logical 
addition and formulae concerning logical multiplication. 

#10*51. h :-~{fax) . <f>x . yfrx] . = :<f>x. D x .~yfrx 
Bern. 
h . #10-252 . D h :.~ {fax) . </># . yfrx] . = :(x) .~(<f>x . yfrx) : 
[#4*51*62.*10*271] = :(x):<f)X.D.^yfrx:.Dh.Yrop 

#10*52. b :. fax) . (f>x . D : (x) . <f>x Dp . = .p 

Dem. 

h . #5*5 . D H :: Hp . D :.p . = : fax) .<f>x.D.p: 
[#10*23] = : (x) . <f>x Dp : : } h . Prop 

#10*53. I- :. ~fax) . <j>x . D : <f>x . D x . yfrx 

Dem. 

h. #2*21. #10*11.3 

h :. (x) :. o-xfrx . D : <f>x . D . yfrx :. 

[#10*27] Dh:.(x).^<f>x.D:(x):<f>x.D.yfrxi. 

[#10*252] D h :.~(aa;) . <f>x . D : (x) : <f>x . D . yfr x :. D h . Prop 

#10*541. \-:.<f>y . D y .pvyfry : = :p. v ,<f>yD y yfry 
Dem. 



h . #4*2 . (#101) .Db:.<f>y.D y .pvyfry: = 
[Assoc.*10*27l] = 

[#10*2] = 

[(#1*01)] 



(y).~<f>yvpvyfry: 

(y).pv~<f>yvyfry: 

p.v.(y).~<f>yvyfry: 

p. v. <f>yD y yfry:.Db. Fro? 



The above proposition is only needed in order to lead to the following: 
10*542. b:.<l>y.D y .pDyfry: = :p.D.<f>yD y yfry f*10*541-^l 
This proposition is a lemma for #84*43. 



*10'55. h :. fax) .<f>x.yfrx:<j>xD x yfrx: = : fax) .<f>x:<f>xD x yfrx 
Dem. 

I- . #4*71 . D I- :. <•&# D yfrx . D : <f>x . yfrx . = . <f>x 

h.(l).*1011'27.D 

I- :. $x D x yfrx .D:(x):<f)X .yfrx . = . <f>x : 

[#10*281] D : fax) . <fix . yfrx . = . fax) . <f>x 

b. (2). #5*32. Dr. Prop 
This proposition is a lemma for #117 , 12 , 121. 



(1) 



(2) 



150 MATHEMATICAL LOGIC [PART 1 

#10'56. I- :. <j)x . D x . yjrx: (g#). $x . X x: ^ -(3^) • ty x 'X x 

Dem. 

h . #10'31 . D h :. <f>x . D x . yfrx : D : cf>x . yx . D x . -tyx . yx : 

[#10*28] D : (ga;) . <£# . %a; . D . (gar) . fa; . ^a; (1) 

h . (1) . Imp . D h . Prop 

This proposition and #1057 are used in the theory of series (Part V). 
#1057. \- :.<f>x .D x . yjrx v yx : D : <f>x Z> x y\rx . v . (g#) . fac . yx 

Dem. 
h . *10-51 . Fact . D 

I- :. <f>x . Da, . fa; v ^a; : ~(g#) . <f>x . yx : "D : <f>x . D x . yjrx v yx : <f>x . D x . <^yx '• 
[*10*29] D :<f>x .D x . fa; v yx . ~ yx : 

[*5'61] D : <£a; . D* . fa; (1) 

h.(l).*5-6.Dh.Prop 



#11. THEORY OF TWO APPARENT VARIABLES 

Summary of #11. 

In this number, the propositions proved for one variable in *10 are to be 
extended to two variables, with the addition of a few propositions having no 
analogues for one variable, such as *ll'2-21-23-24 and *ll-53-55'6-7. "<f> (x, y) 
stands for a proposition containing x and containing y\ when x and y are un- 
assigned, <f> {x, y) is a prepositional function of x and y. The definition #1101 
shows that " the truth of all values of <f> (x, y)" does not need to be taken as a 
new primitive idea, but is definable in terms of " the truth of all values of yfrx." 
The reason is that, when x is assigned, <f>(x,y) becomes a function of one 
variable, namely. y, whence it follows that, for every possible value of x, 
"(y).4>(x,y)" embodies merely the primitive idea introduced in *9. But 
"(y) . <f> (x y y)" is again only a function of one variable, namely x, since y has 
here become an apparent variable. Hence the definition #1101 below il- 
legitimate. We put: 

#1101. (x,y).<j>(x,y). = \(x):(y).<i>(x,y) Df 

#1102. (x, y, z). <J> Qc, y, z) . = : (x) : (y, z) . <f> (*, y, z) Df 

*n-03. (aa?,y).^(a?,y). = :(a*):(ay)-^(^y). Df 

*ll-04. (^x,y,z).<f>(x,y,z).^:(^x):('3_y,z).<f>(x,y,z) Df 

*ll-05. <j>(x > y).D x>y .1r(x,y)i = i(x,y):<j>{x,y).3.+(x,y) Df 
*ll-06. ^(a?,y).s,, y .^(«,y): = :(«.y)!*( aj .y)-=-^ , ( aj 'y) Df 
All the above definitions are supposed extended to any number of variables 
that may occur. 

The propositions of this section can all be extended to any finite number 
of variables; as the analogy is exact, it is not necessary to carry the process 
beyond two variables in our proofs. 

In addition to the definition #11-01, we need the primitive proposition 
that "whatever possible argument x may be, <f> (x, y) is true whatever possible 
argument y may be" implies the corresponding statement with x and y inter- 
changed except in "<f>(x,y)". Either may be taken as the meaning of 
"<£0, y) is true whatever possible arguments x and y may be." 

The propositions of the present number are somewhat less used than those 
of #10, but some of them are used frequently. Such are the following: 
#111. \-:(x,y).(f>(x,y).D.<f>(z,w) 

#11-11. If <f> (z, w) is true whatever possible arguments z and w may be, then 
(x, y) . <j> (x, y) is true 

These two propositions are the analogues of #10111. 



152 MATHEMATICAL LOGIC [PART I 

*ll-2. H:tey).*tey). = .(y f *).$(*,y) 

I.e. to say that "for all possible values of x, <j>(x, y) is true for ail possible 
values of y" is equivalent to saying "for all possible values of y,<f>(x,y) is 
true for all possible values of x." 
*11'3. f- : . p . D . (x, y) . <f> (x, y) :=:(x, y) : p . D . <f> (x, y) 

This is the analogue of *1021. 
*1132. \-:.(x, y) : <f> (x, y>. D . ^ (x, y) : O : (x, y) . <f> {x,y) . D .,(*, y) . ^ (x, y) 
I.e. "if <f>(x,y) always implies y{r(x,y), then '<f>(x, y) always' implies 
'f (®> y) always.'" This is the analogue of *1027. *ll-33-34341 are respec- 
tively the analogues of *10-27r28'281, and are also much used. 
*ll-35. r :.(x,y) : <f>(x, y) . D .p : = : fax, y) . <f> (x,y) .D.p 

I.e. if <f> (x, y) always implies p, then if <f> (x, y) is ever true, p is true, and 
vice versa. This is the analogue of *1023. 
*ll-46. t-:.(&x,y):p.<t>(x,y): = :p:(>&x,y).<f>(x,y) 

This is the analogue of *1 0'35. 
*H54. ^:.faso,y).<f>x.ylry.^:fax).<f>x:fay).yfry 

This proposition is useful because it analyses a proposition containing 
two apparent variables into two propositions which each contain only one. 
"<j>x.y]ry" is a function of two variables, but is compounded of two functions 
of one variable each. Such a function is like a conic which is two straight 
lines: it may be called an "analysable" function. 
*11'55. r- : . fax, y) . $x . -f (x, y) . = : fax) : <f>x : (ay) . yjr (x, y) 

I.e. to say " there are values of x and y for which <f>x . yjr(x, y) is true " is 
equivalent to saying " there is a value of x for which (f>x is true and for which 
there is a value of y such that yfr (x, y) is true." 
*ll-6. h :: fax) :. fay) . <f> (x, y) . yjry : X x :. = :. fay) :. fax) . <f> (x, y) . X x : ^y 

This gives a transformation which is useful in many proofs. 
*ll-62. h :: <f>x .f(x,y) . 3 x>y . X (x,y): = :. $x . D, : <f>(x, y) . D y . x (x, y) 
This transformation also is often useful. 



*1101. (cc,y).<t>(x,y). = ;{x);{y).<l>{x,y) Df 

*1102. (x,y,z).<\>tx,y,z). = :{x)i{y y z).<\>{x,y,z) Df 

*ll-03. (^,y).<f,(x y y). = :fax):fay).<f>(x,y) Df 

*1104. (^,y,z).<f>{x,y,z). = :fax):fay,z).(f>{x,y,z) Df 

*1105. <l>(x,y),D Xt y.yfr(x > y): = :(x,y):<f>(x,y).^.^(x,y) Df 

*H-06. <f>(x,y).= x ,y.ylr(x,y)t = :(x,y):<f>(x,y). = .,jr(x,y) Df 
with similar definitions for any number of variables. 

*1107. "Whatever possible argument # may be, <f>(x,y) is true whatever 
possible argument y may be " implies the corresponding statement with x and 
y interchanged except in "<f>(x, y)". Pp. 



SECTION B] THEORY OF TWO APPARENT VARIABLES 153 

*H1. h : (x, y) . $ (x, y) . D . (2, w) 

Dem. 

h.*101.DI-:Hp.D.<y).*(*,y). 
OlO'l] O. <£(>,«/) .OK Prop 

*11'11. If </> (z, w) is true whatever possible arguments z and w may be, then 
(x, y) . <f> (x, y) is true. 

Dem. 

By #1011, the hypothesis implies that (y).(f>(z,y) is true whatever 
possible argument z may be; and this, by #101 1, implies (x, y) . <f> (x, y). 

#1112. h :. (x, y) .p v <f> (x, y) . D : p . v . (x, y) . <f> (x, y) 

Dem. 
H.*10'12.Dh:.(y) v pv$tey). Z>:p .v .(y) .<f>{x,y):. 
[*10'll-27]Db:.(x,y).pv<f>(x,y).D:(x):p.v.(y).<l>(x,y): 
[*1012] D : p . v . {x, y) . <f> (x, y) :. D h . Prop 

This proposition is only used for proving #11*2. 

#1113. If <f> (&, y), yfr (&, §) take their first and second arguments respectively 
of the same type, and we have "f- . <j> (x, y)" and "h . \Jr (x, y)" we shall have 
" b .<f>(x, y) . jr (x, yy [Proof as in #1013] 

#1114. Y : . (x, y) . <j> (x, y) : (x, y) . i|r (x, y) : D : <j> (z, w) . yjr (z, w) 
Dem. 

h.*1014.DH.Hp.D:(y).^(^,y):(y).^(^y) 

[#1014] D : </> (z, w) . yfr (z, w) :. D h . Prop 

This proposition, like #1014, is not always significant when its hypothesis 
is true. *1113, on the contrary, is always significant when its hypothesis is 
true. For this reason. #1113 may always be safely used in inference, whereas 
#1114 can only be used in inference {i.e. for the actual assertion of the con- 
clusion when the hypothesis is asserted) if it is known that the conclusion is 
significant. 

#11-2. I- : (as, y) . <f> (x, y) . = . (y, x) . <fj (x, y) 
Dem. 

V . #11-1 . D h : {x, y) . <f> (x, y).D.<f>(z, w) (I) 

V . (1) . #11-0711 . D r :. (w, z) : (x, y) . <j> (x, y) . D . <f> (z, w) (2) 

V 

h :. (x, y).(f>(x,y).D. (w, z) ,<f>(z, w) (3) 

Similarly I- : . (w, z) . <f> (z, w) . D . (x, y) . <f> (x, y) (4) 

I- . (3) . (4) . D H . Prop 

Note that "(w, z) . <f> (z, w)" is the same proposition as " (y, x) . <f> (x, y)"; 
a proposition is not a function of any apparent variable which occurs in it. 



154 MATHEMATICAL LOGIC [PART I 

♦1121. I- : (x, y, z) .(f>(x,y,z). = . (y, z 3 x).$ (x, y, z) 
Dem. 



.(x):.(y):(z).<f>(x,y,z):. 
.(y):.(x):(z).<f>(x,y,z):. 
.(jj)i.(z)\(x).4>(x;y,z)u 
. (y, z, x) . <f> (x, y, z) :: D b . Prop 



[(♦11-01-02)] I- :: (*, y r z) . $ (x, y,z).= 
[*ll-2] 

[*11-2.*1 0*271] = 

[(♦H'01-02)] = 

♦1122. b:(<Rx,y).<f>(x,y). = .~{(x,y).~(f>(x,y)} 
Dem. 

b . ♦10252 . Transp . (*ir03) . D 

b:(^x,y).<f>(x,y). =.~i(*):~(ay).^(a?,2/)} ■ 

[♦10252-271] s . ~ {(x) : (y) . ~ <£ (x, y)\ . 

[(♦11-01)] = • ~ {(*, y) • ~ (*. y)) • 3 *" • Prop 

♦H'23. b:(^x i y). < f>(x,y). = .( l 3_y,x).<j>(x,y) 

Dem. 

b . *11'22 .Db: K -£<c, y) . <j>{x, y) . = . ~ {(#, y) . ~ <*> (x, y)} . 

[*ll-2.Transp] = • ~ {(y, #) ■ ~ <j> (x, y)} • 

[♦11-22] = • (a#> x ) ><f>(x,y):3b. Prop 

♦11-24. i-:(a«,y,^).^(^y,^). = .(a2/,^«).^(^y./) 

Dem. 
[(♦11-03-04)] b :: (gar, y,z).<f> (x, y,z). = :. (a*) :. (ay) : (a*) .<£(#, y, *) :. 

[♦ii-23] = - (ay) ». (a* 7 ) ■• (a*) • 4> (®> &>*)'•• 

[*ii-23.^io-28i] =:-(ay):.(a^):(a*)-^(«,y,«)» 

[(♦11-03-04)] = :. (ay, *,«).£(*,&*):: 3 K Prop 

♦11-25. b:~{(<&x,y).<l>(x,y)}. = .(x,y).~<l>(x,y) [*1 122 .Transp] 
♦11-26. r- : . (a«) : (y) . £ (a?, y) : 3 : (y) : (a»> • 4> (*» 2/) 

b . ♦101-28 . D b :. -(a*) : (y) . <£ (a?, y) : > : (a*) ■ <£ 0*. ^) C 1 ) 

I- . (1) . ♦10-11-21 . D b . Prop 
Note that the converse of this proposition is false.. E.g. let <j>(x,y) be the 
propositional function " if y is a proper fraction, then x is a proper fraction 
greater than y." Then for all values of y we have (a#) • <£ («, yX s0 tnafc 
(y) : (3*) ■ 4* ( x > V) is sa ti8ned - In fact ' (y) '• (ft®) • <t> ( x > $)" expresses the 
proposition: " If y is a proper fraction, then there is always a proper fraction 
greater than y." But "(&x) : (y) . <£ (x, y)" expresses the proposition: " There 
is a proper fraction which is greater than any proper fraction," which is 
false. 

♦11-27. b :.(a*,y) •{ T Kz)-^(x,y;z)' s? : (a*) * to *) ■ ^ (*> & *> = 

= '{'&x > y,z).<$>(x,y,z) 



SECTION B] THEORY OF TWO APPARENT VARIABLES 155 

Bern. 

K #4*2. (#11*03). 3 

i- - (a*> y) = (a^) • 4>i x > y> z ) • = •■ (a#> -(ay) : (a*) ■ 4> {*> y> *> (*) 

K*4*2.(#ll*03).O 

h : • (ay) = (a- 2 ) ■ <f> («*> y> z )- = - (ay* *) • #(^ u> z ) ( 2 ) 

f-r.,(2) . *10*11-281 . y 

h ::-(a») :. (ay) : (a?y.. <£ o», y^-- = "---(a*) r (ay> *) • f (*>y» ■*) ( 3 ) 

H . (1);..(3) . (#11*04) . D h . Prop 
All the propositions of #10 have analogues whichi hold for two or more 
variables. The more important of these are proved in what follows. 

*ll-3, bz.p.O. (x,y) . <fi (x, y) r= : (x,y) : p *D-4> (x, y) 

Dem. 
b . #10-21. >h :.p . D . (ayy) . 4>{x,y) : = i(x) :p~D .(y) . ^(«, y) : 
[#1 0-21-271] = : (x, y) :p . D . <£ (*,.y>:.. > I" ■ Prop 

*11'31. I- : . (x, y) . <j> (a; , y) : (a>, y) . f (x, y) : = : (x, y) : $ (x x y) . ty (#, y) 

Here the conditions of significance on the right-hand side require that 
<f> and ^should take arguments of the same types. 
Dem. - 

h.*l0'22.Dh::(x,y).<]>(x,yy.(x,y).ylr(x,y): 

= :.. • («■) :■ (y) *<£ (*, y)r(y) ."f (*, y) i. 
[*10*22*271] = :. (*, y) :</>(*, y>.*foy) K D b . Prop 

The proofs of most of the following propositions are conducted exactly as 
those of #11*3*31 are conducted: the analogous proposition in #10 is used 
twice, together with #10"27 or #1 0*271 or #10*28 or #10*281 as the case may 
be. When proofs conform to this pattern we shall merely give references to 
the propositions used. 

#11*3.11. If <f> (£,§), ■\jr(^,p) take arguments of the same type, and we have 
" b . <f> (x, y)" and " b . yfr (x, y)," we shall have " b . $ (x, y) . f (x, y)." [Proof 
as in #10*13.] 
#11*32. b:.(x,y):<f)(x > y).D.^ (x, y) : D : (x, y). <f>(x,y).D. (x, y) . yjr (x, y) 

[#10-27] 
*11*33. b :.(x, y) : <f> {x, y). = .ty (x,y) z D : (x, y) . <f> (x, y) . = . (x, y) . -f (x t y) 

[#10*271] 
#11*34: b :. (x,y) : <f> (x, y) . 3 . ^ («, y) : 3 = 

(^x,y).4>(x,y).D.(^x r y)..f(x,y) [*10^7*28],< 
#11*341. h :. (x, y) : <f> {x, y) . = . ^ (x, y) : D : 

(rx, y).<f>(x,y). = . (rx, y) . ^ (x y y) [*10-27l*281] 
#11*35. b:.(x, y) : <j> (x, y) . D . p : = : (rx, y) . cf> (x, y) . 3 . p [*10*23*271] 
#11*36. b : <f> (z, w) . D . (rx, y) . (x, y) 
Dem. 

b . #1 1*1 . D h : (x, y) . ~> <f> (x, y) . D \ ~ <f> (z, w) (1 ) 

|-.(l).Transp.DI-.Prop 



1^6 MATHEMATICAL LOGIC [PART I 

*11'37. ^•.•■(x,y):(f>(x,y).D.yJr(x,y):.(x,y):yJr(x,y).D. x (x ) y):. 

3:(x,y):<f>(x,y).3. x (x,y) 
Dem. 

In the following demonstration, " Hp" means the hypothesis of the propo- 
sition to be proved. We shall employ this abbreviation, whenever convenient, 
in all cases where the proposition to be proved is a hypothetical, i.e. is of the 
form "p3q." Similarly "Hp (1)" will mean "the hypothesis of (1)," and 
so on. 

H.*ll'Sl.Dh::Hp.D:.(*,y):.^(«,y).D.^(*,y):^(^y).D. X (* f y) (1) 

»--Syll.*ll-ll.Dh:'.(« > y):.^(«,y).D.^(*;y).:Vr(*,y).D.x(*.y): 

2:<f>(x,y).D. x (x,y):. 

[*ll-32] D\-:.(x,y):<f>(x,y).D.\jr(x,y):^(x,y).D. x (x,y): 

l /n /ox a „M „ ^:(x,y):<f>(x,y).D. X (x,y) (2) 

K (1) . (2) . Syll . D h . Prop 

The above is a type of proof which recurs frequently in what follows. 
Proofs conforming to this pattern will be indicated only by the numbers of 
the propositions used. 

*11'371. I- :: (a?, y): ^(x,y). = .^ (x, y) :. (x, y) : -f (x, y) . = . x (as, y) :. 

3:.(x,y):<f>(x,y). = . x (x,y) [*U-3111-33] 
*1138. \-::(x,y):<f>(x,y).D.ylr(x,y):.l:. 

(x,y):<f)(x,y). X (x > y).^.y(r(x,y). X (x,y) [Fact .*1 111 "32] 

*11'39. h::(x,y):<)>(x > y).^.yJr(x i y):.(x,y)i X (x ) y).^.0(x,y):.D:. 
(x, y) : $ (x, y) . X (x, y) . D . -f (x, y) . (x, y) [*3'47 . *1111'32] 

*11391. h :: {x, y) : <f>(x, y) . D . ^ (x, y) :. {x, y) z <)> (x, y) . D . x (x, y) :. 

= :(x > y):<f>(x,y).D.yjr(x,y). x (x,y) 
Dem. 

H.*4-76. 3h:.<j>(x,y).0.+(x,y):<f>(x t y).D. x (x,y): 

= :<t>(x,y).D.yfr(x,y).x (x, y) :. 

[♦11-11-33] •Dh:.(x y y):<f>(x,y).D.yJr(x,y):<l>(x i y).D. x (x,y): 

= ' &> V) :<}>(x > y).O.yf r (x, y) . X {x,y):: 

[*11-31] ^^:-(x ) y):<j>(x,y).D.f(x,y):.(x ) y):(f>{x > y).D. x (x,y):. 

= i(x,y):<f>(x,y).D.'f(x > y). x (x,y):: 
D h . Prop 

*ll-4. > :: (x, y) : <f> (x, y) . = . yfr (x,y) :. (x, y) i X (x, y). = .0(x, y) :. D :. 
(x,y):<f>(x,y). x (x,y). = .yjr(x > y).0(x,y) 
Dem. 

^.^llSl.Dh::Hi ? .D:.(x > y):.<f>(x > y). = ^(x ) y): X (x ) y). = :0(x,y):. 
[*4-38.*ll-ll-32] 3:.(x,y):<f>(x,y). x (x y y). = .ylr(x,y).e(x,y):: 

I- . Prop 



r*ll-4^.Id] 



SECTION B] THEORY OP TWO APPARENT VARIABLES 157 

#11401. b::(x,y):<f>(x,y). = .yjr(x,y):-D:. 

(x, y) : 4> (x, y) . x (#, y) ■ = • -f («, y) • X ( x > V) 
*11'41. I- :. fax, y).<j>(x,y):w: fax, y) . ^ (x, y) : 

= :fax,y):<f>(x,y).v.yjr(x,y) [*10-42'281] 

*11'42. b :. fax, y) . <f> (x, y) . yjr (x, y) . D : fax, y) . <f> (x, y) : fax, y) . yfr (x, y) 

[*10-5] 
#11-421. h :. (x, y) .<f>(x,y).v. (x, y) .yjr(x,y):0 : (x, y) : <f> (x, y) . v . yjr (x, y) 

*ll-42 ^J*'^ • Transp . #456] 

*11'43. h : . fax, y):<f>(x,y).D.p: = :(x,y).<f>(x,y).D.p [#10'34-281] 

#11-44. h : . (x, y) : <f> (x, y) . v . p : = : (x, y) . <f> (x, y) . v . p [*10-2271] 

*11'45. H :. fax, y):p.4>{x,y): = :pifax,y).<\> (x, y) [*10-35281] 

#11-46. H :. fax, y):p.3.<t>(x,y): = :p.D. fax, y) . <j> (x,y) [*10-37'281] 

*11'47. h :. (x, y) :p . <f> (x, y) : = : p : (x, y) . <f> (x, y) [*1033-27l] 

*ll-5. V :. fax) :~{(y) . <f>(x,y)} : = :~{(x,y) . (f>(x,y)} : = : fax,y) .~<f>(x,y) 
Bern. 
■ h.*lO-2o2.D)-:.fax):~{(y).<l>(x,y)}:=:~{(x):(y):<l>(x,y)}: 

[(*n-oi)] = :-{(*, y)-*(*,y)} (i) 

I- .#10-253 . D I- :~{(y) . <f>(x, y)} . = . fay).~<f>(x, y) : 

[(#11-03)] ='-fax>y)-~4>(%,y) (2) 

K(l).(2). DKProp 

#11-51. h : . fax) :(y).(f>(x,y): = :~ {(x) : fay) . ~ <f> (x, y)} 

Dem. 
V . #10-252 . Transp . D V : . fax) : (y) . <f> (x, y) : = : ~ [(x) : ~ (y) . <f> (x, y)] (1 ) 
I- . #10253 .Dh:.~(y).<f>(x,y). = : fay) . ~ <f> (x, y) :. 

[#10-11-271] D h :. (x) :~(y) . (#, y) : = : (a;) : fay) .~<f> (x, y) :. 

[Transp] D h :.~[(*) :~{(y)- ^(^y)0- = =~{(«) = (3y).~0(*,y)} (2) 
h . (1) . (2) . D h . Prop 

#11-52. H :. fax, y) . <f> (x, y) . yfr (x, y) . = .~{(#, y) :<j>(x,y).D .~yjr (x, y)} 
Dem. 

h . *4-51-62 . D 

h:.~\<f>(x,y).\lr(x,y)}. = :<f>(x,y) .D .~yjr(x,y) (1) 

H.(l). #1111-33. D 

l-:.(^,y) — {<f>(x,y).yJr(x,y)}: = :(x,y):<f>(x,y).D.~y}r(x,y) (2) 

I- : (2) . Transp . *1122 . D h . Prop 

#11*521. H :. ~fax, y) . $ (x, y) . ~f (x, y) . = : (x, y) : <f> (x, y) . D . ^ (x, y) 

>^r(x,y) 



#11 -52. Transp. 



yjr(x,y) 



158 MATHEM&TI'CAIi LOGIC [PART I 

#11*53. 4-:. t>, y). 0aO<fy. =? :(!«#. 0a? . ^ -%) ■ "^ 

Dem. 
1- . *10-21-271 . D1-*.(sg, if}*if><v3fy ..■= : (a?) : 0a?. D . (y).. -^t/.: 
[#1023] =r(a«).<^.D.(2/).^:.DH.Prop 

*11'54. I- :. <ga?, y).. 0a; . i/ry . = : (ga;) . 0a? : (gy) . fy 

Dem. 
t- . #1035. 3 h :. (33/) . 0a; . ijry . ==:: 0a? : (gy) ■ ^ y :■ 
1*1011 -281 ] 3 h : . (ga>, y) . jfxc . -^y .. = : (gar) : 0a; : (gy) .. ^y : 
f*10-35] =.: (gar) .0a?: (gy) . -fy :.3> .Prop 

This proposition is very often used. 

#11-55. h :. (gas, y) • 0a? • ^(a?, #)- = : (g«?) : 0a; : (gy) . ijr (a;, y) 

Dem. 
f- . #1035 . D V : . (gy ) . 0a; . yjr (w,, y) . = : $x : (gy) . f (a?, y ) : . 

£#1011] D h :. (a?) 1. (gy) .-.0a; . fix, y) . = : (f>x : (gy) . ^ (a?, y) :. 
£#10-281] D f- :.<gaj) : (gy),. 0a? . *jr (x, y).= : (ga?)i 0a? : (gy) . f(x,y) :. D h . Prop 

This proposition is very often used. 

#11 56. H :. (x)~<j>x : (y) . tyy-. s- : .(*, y) . *0a? . -»|ry 
Dera. 

> .#10-33 . D H :: (a?) .-. 0a; : (y) . ^y : = :. (a?) :. 0a; : (y) . f y (1) 

K'*10'38.D.r-:. <j>x :(y) . fy : = : (y). <f>x.fy :. 

[*10'113 D h :. (a;) :.0a; : (y) . ip»y : = : (y) .<f>x.fy:. 
[#10-271] > h::(x) :.xf>x: (y) . -0-y :. = : (a?) : (y) . <f>x . yfry : 
.[(#11-01)]' = :(x,y).<f>x.fy (2) 

h . (1) . (2) . D H . Prop 

#11-57. H : (x) . 0a; . = . (x, y) . 0a>- ^y £*T1S6 . #4-24] 

The use of *4'24 here depends upon the fact that (a;) . 0a; and (y) . 0y are 
the same proposition. 

#11-58. r- : (ga;) . 0a--, = . (ga>, y) . 0a; .0y £#11 -54. #4-24] 

*ll-59. > : . 0a; . D x . sfrx : = z (f>x . 0y . D*, y .^x.y\ry 
Bern. 
h .#11*57 . D f:s. 0a; . 3^ • ty® ' = ' (a?, y) : (f>x . D . yjrx : (f>y . D . yjry : 
[*3-47.*ll-32] 3 r <a?, y) : 0a; . 0y . D .^x.tyy (1) 

1- . #11-1 . D> :. {x, y) : <f>x . (f>y . D . tyx .. -ty-y :D : $x . 0y . D . i/ra; . -^-y (2) 
[- . ( 2 ) - . *4-24 . 3 h :. Hp (2) . D : 0a; . D . . y}rx (3) 

f. (3) . #1011-21 . D 

h :. (x, y) : 0a; . 0y . D - -frx . -^ry : D : 0a; . D^ . ^|ra; (4) 

l-.(l).(4).Dh.Prop 



SECTION B] THEORY OF TWO APPARENT VARIABLES 159 

*ll-6. I- :: (g«) :. (ay) . <f>(x, y)-^y- X Xi - s : * (32/) '• (3*) • $(*• V) • X x • t$ 
This proposition is very frequently employed in subsequent proofe. 
Dem. 

h.*10*35. D hi. (%y). fix, y).ylry:xx'. = :(fty) -.fix* V)-.+y •%*'•• 
[*10-11-281] D I- :: (a«) :■ (32/) • <f> («, 3/) -*y-X xi 

= :. (a«) :. (ay) . £ (#, y) . yjry . X x '.. 

[*n-23] = :. (ay) '• (a*) ■ 4> (x> y) • -irv ■ x* : - 

[*1 1 '341 .Perm] = : . (ay) : • (a* 7 ) ■ $ (*. y)'X x -^ry- 

[fcio-35-2813 = :. (ay) *- (a#) ■ ^ (** y)'X x - "fy w3+- -Rop 

•11-61. I- i. (ay) :<f>x.D x .-4r (x, y) : D : <f>x . D x . (ay) - "f («vy) 

Dem. 
h . *ll-26 . D r :.: Hp . D :. (a?) :. (ay) : ty.O . + (x, y) (1) 

V . *1037 . D r- :. (ay) : <£* . D . ^-(a;, y) : D : <fce . 3 . (3y) ■ lK*» y) » 
[*10-11'27]D I- ::.<«) :.<ay) : ^ • ^ ■ * <*. y) :.:>:.(*):**. 3. (ay). *(*,y) (2) 
h.(l).(2)OI-.Prop 

*11;62. I- :: <f>x . ^ (x, y) . x>y . % (a?, y) : = :. <f>x . D* : yjr (x, y) . 3 y . x(a?,y) 

h.*4-87.*llll-33.D 

I- :: <£# . yfr (x, y) . D XiV . x (x, y) : = :• (#, y) :. £# . D : ^r(^, y) . D . # (#, y) 

[*10-21'11-271] s:.(*)r.^O:(y)*f (« f y).3.*(«vy*« 

DKProp 
•1163. h : . ~ (a«, y) . £■(*, y).Oi<f>(x, y) - D., „ . ^ (x, y) 

Dem. 
r- . *2-21 . *1111 . D I- :. (x, y) i.~<f>(x,y) . D : <f>(x, y) . >,^r(x, y)u 
[*1I'32] D I- :. (x, y) ~<*»<f>(x, y) . D : (a?, y) : <£ (x, y).1.^{x,y) :. 

[•11-25] 3^:-~(a«»y)-*<*,y)-3s-<*,y)«^<*,y).3.^(*,yM- 

Dh.Prqp 
•11-7. J- : . (a«, y) : <f> (ayy) . v . £ (y, ar) : = . (a#, y) . $ (x, y) 
Dem. 

V . *1 1 41 . 3 h :. (a*, y) : ^ (^, y) • v . #<y, ar) : 

= : (3*>y) • $ (*, y) • v . (a*,.y>. ^><y, x) : 

C* 11 ' 23 ] =:(a^y)-<^(^y)-v.(ay,^).0(y,a;): 

[*4'25] 5:(30,y)*^-(*>y):-3Kfrop 

In the last line of the above proo^ use is made of the fact that 

(a«» y) • 4> (*» y) a 1 " 1 <ay» *) • i> (y, ®) 

are the same proposition. 

The first use of the following proposition occurs in the proof of *234'12. 
Its utility lies in its enabling us to pass from a hypothesis 

containing two apparent variables, to the product of two hypotheses each 
containing only one. 



160 MATHEMATICAL LOGIC [PART I 

#11-71. h :: (qz) . fa : (aw) . %w : D :. 

fa . D z . -\Jrz : %w . D w . 6w : = : fa . xw . D z>w .yfrz.Bw 
Dem. 

V . #101 . #347 . D b :. fa . D z . yfrz : %w • X • Bw : 

D : fa . yw . D . yjrz . &w (1) 
h . (1) . #1111-3 . "D\-:.fa.D z .fz: X w.D w .0w: 

3 : fa . "XW . ZtW . yfrz . 6w (2) 

V . #101 .Dhr.fa. x w • "^z, w • ^ z ■ Bw : D :. fa . %«/ .D w .yfrz ,0w:. 

[#10-28] D :. (gw) .fa.xw.D. (gw) . ^vs . #w :. 

[#1035] D :.fa:(ftw).xw:D lylrzifawy.Bw 

(3) 
h . (3) . Coram . #3*26 . D h :: (gtt>) . x w '^ '•■ <l> z • X w ■ ■}*,» • tyz .0w: 

Dzfa .0 .yfrz (4) 
I- . (4) . #10-11-21 . D I- :: (gw) . %w . D :. ^ . %w . D Z)W l . ^ . 0w : 

2 : fa . D z . tyz (5) 

Similarly h :: (qz) .fa.Di.fa. \w . D 2)W . ^ . Ow : 

D : %?# .D w .0w (6) 
H . (5) . (6) . #3-47 . Comp . D 

h :: Hp . D :'. fa . %w . D z>w . ^ . 0w : D : <£z . D z . \|r^ : ^ty ,"2> w .0w (7) 

h.(2).(7).DKProp 



*12. THE HIERARCHY OF TYPES AND THE AXIOM 
OF REDUCTIBILITY 

The primitive idea "(x) . <$>x" has been explained to mean "<f>x is always 
true," i.e. "all values of <j>x are true." But whatever function <f> may be, there 
will be arguments x with which <f>x is meaningless, i.e. with which as argu- 
ments $ does not have any value. The arguments with which <\>x has values 
form what we will call the "range of significance" of cf>x. A "type" is defined 
as the range of significance of some function. In virtue of *9*14, if <j>x, <f>y, 
and yjrx are significant, i.e. either true or false, so is yfry. From this it follows 
that two types which have a common member coincide, and that two different 
types are mutually exclusive. Any proposition of the form (x) . cf>x, i.e. any 
proposition containing an apparent variable, determines some type as the 
range of the apparent variable, the type being fixed by the function <j>. 

The division of objects into, types is necessitated by the vicious-circle 
fallacies which otherwise arise*. These fallacies show that there must be 
no totalities which, if legitimate, would contain members defined in terms of 
themselves. Hence any expression containing an apparent variable must not 
be in the range of that variable, i.e. must belong to a different type. Thus 
the apparent variables contained or presupposed in an expression are what 
determines its type. This is the guiding principle in what follows. 

As explained in #9, propositions containing variables are generated from 
propositional functions which do not contain these apparent variables, by the 
process of asserting all or some values of such functions. Suppose <f>a is a 
proposition containing a; we will give the name of generalization to the 
process which turns <f>a into (x) . <j>x or (ga?) . <f>x, and we will give the name 
of generalized propositions to all such as contain apparent variables. . It is 
plain that propositions containing apparent variables presuppose others not 
containing apparent variables, from which they can be derived by generaliza- 
tion. Propositions which contain no apparent variables we call elementary 
propositions^, and the terms of such propositions, other than functions, we call 
individuals. Then individuals form the first type. 

It is unnecessary, in practice, to know what objects belong to the lowest 
type, or even whether the lowest type of variable occurring in a. given context 
is that of individuals or some other. For in practice only the relative types 
of variables are relevant; thus the lowest type occurring in a given context 
may be called that of individuals, so far as that context is concerned. Accord- 
ingly the above account of individuals is not essential to the truth of what 

* Cf. Introduction, Chapter II. 
t Cf. pp. 91, 92. 
R&W I 11 



162 MATHEMATICAL LOGIC [PART I 

follows; all that is essential is the way in which other types are generated 
from individuals, however the type of individuals may be constituted. 

By applying the process of generalization to individuals occurring in 
elementary propositions, we obtain new propositions. The legitimacy of this 
process requires only that no individuals should be propositions. That this is 
so, is to be secured by the meaning we give to the word individual. We may 
explain an individual as something which exists on its own account; it is then 
obviously not a proposition, since propositions, as explained in Chapter II of 
the Introduction (p. 43), are incomplete symbols, having no meaning except 
in use. Hence in applying the process of generalization to individuals we run 
no risk of incurring reflexive fallacies. We will give the name of first-order 
propositions to such as contain one or more apparent variables whose possible 
values are individuals, but contain no other apparent variables. First-order 
propositions are not all of the same type, since, as was explained in *9, two 
propositions which do not contain the same number of apparent variables 
cannot be of the same type. But owing to the systematic ambiguity of nega- 
tion and disjunction, their differences of type may usually be ignored in practice. 
No reflexive fallacies will result, since no first-order proposition involves any 
totality except that of individuals. 

Let us denote by "<f> ! x" or "<f> ! (&, §)" or etc. an elementary function whose 
argument or arguments are individual. We will call such a function a, predi- 
cative function of an individual. Such functions, together with those derived 
from them by generalization, will be called first-order functions. In practice 
we may without risk of reflexive fallacies treat first-order functions as a type, 
since the only totality they involve is that of individuals, and, by means of the 
systematic ambiguity of negation and disjunction, any function of a first-order 
function which will concern us will be significant whatever first-order function 
is taken as argument, provided the right meanings are given to the negations 
and disjunctions involved. 

For the sake of clearness, we will repeat in somewhat different terms our 
account of what is meant by a first-order function. Let us give the name of 
matrix to any function, of however many variables, which does not involve any 
apparent variables. Then any possible function other than a matrix is derived 
from a matrix by means of generalization, i.e. by considering the proposition 
which asserts that the function in question is true with all possible values or 
with some value of one of the arguments, the other argument or arguments 
remaining undetermined. Thus e.g. from the function <f> (x, y) we shall be able 
to derive the four functions 

(x).<f>(x,y), (a#).<£0, y)> {y)>4>( x >y)> (ay)-<M^y)> 

of which the two first are functions of y, while the two last are functions of x. 
(All propositions, with the exception of such as are values of matrices, are also 
derived from matrices by the above process of generalization. In order to obtain 



SECTION B] THE AXIOM OF REDUCIBILITY 163 

a proposition from a matrix containing n variables, without assigning values 
to any of the variables, it is necessary to turn all the variables into apparent 
variables. Thus if <j> (x, y) is a matrix, (x, y) . <f> (x, y) is a proposition.) We 
will give the name first-order matrices to such as have only individuals for 
their arguments, and we will give the name of first-order functions (of any 
number of variables) to such as either are first-order matrices or are derived 
from first-order matrices by generalization applied to some (not all) of the 
arguments to such matrices. First-order propositions will be such as result 
from applying generalization to all the arguments to a first-order matrix. 

As we have already stated, the notation "<j> ! z" is used for any elementary 
function of one variable. Thus "(f) ! x" represents any value of any elementary 
function of one variable. It will be seen that "<j)lx" is a function of two 
variables, namely <j> ! z and x. Since it contains no apparent variable, it is 
a matrix, but since it contains a variable (namely (f> I z) which is not an in- 
dividual, it is not a first-order matrix. The same applies to ! a, where a is 
some definite constant. We can build up a number of new matrices, such as 
~<jE>!a, ~<f>lx, (fylxv (f>ly, fylxvtylx, (f>lxvyfrly, 
<f>lx . D .y\r\x, Qlx.-^rlx, <|)!a;vf !yv^!0, and so on. 
All these are matrices which involve first-order functions among their argu- 
ments. Such matrices we will call second-order matrices. From these matrices, 
by applying generalization to their arguments, whether to such as are functions 
or to such (if any) as are individuals, we obtain new functions and propositions. 
Such functions (together with second-order matrices) will be called second- 
order functions, and such propositions will be called second-order propositions. 
Thus we are led to the following definitions: 

A second-ord&r matrix is one which has at least one first-order matrix 
among its arguments, but has no arguments other than first-order matrices 
and individuals. 

A second-order function is one which either is a second-order matrix or 
results from one by applying generalization to some (not all) of the arguments 
to a second-order matrix. 

A second-order proposition is one which results from a second-order matrix 
°y a PPly m g generalization to all its arguments. 

In addition to the above illustrations of second-order matrices, we may 
give the following examples of second-order functions : 

(1) Functions in which the argument is $ ! z : (x) . <£ ! x, (g#) . <f> ! x, 
<f>la.D.<f>lb, where a and b are constants, <f> I x . D x . g ! x, where g ! z is a 
constant function, and so on. 

(2) Functions in which the arguments are ! z and yjrlz: 
<f>lx.D x .yfrlx, <f)lx.= x .ylrlx, (ga?) .<j>x.-fx, <f> ! a . D . -f ! b, 

where a and b are constants, and so on 



11—2 



164 MATHEMATICAL LOGIC [PART I 

(3) Functions in which the argument is an individual x : (<f>) .<j>loc, 
(3$) •<£!#,<£!#• 3$ ■ <M a, where a is constant, and so on. 

(4) Functions in which the arguments are <f> I z and x: $!#,<£!#. D .<£! a, 
where a is constant, (g/»/r) : <£ ! x . = . yjr ! x, and so on. 

Examples of second-order functions might, of course, be multiplied in- 
definitely, but the above seem sufficient for purposes of illustration. 

A second-order matrix of one variable will be called a predicative second- 
order function of one variable or a, predicative function of a first-order matrix. 
Thus ! a, ~ <f> ! a and <f> ! a D <j> ! b are predicative functions of (f> ! z . Similarly 
a function of several variables of which at least one is a first-order matrix, 
while the rest are either individuals or first-order matrices, will be called 
predicative if it is a matrix. 

It will be seen, however, that a second-order function may have only 
individuals for its arguments; instances were given just now under the 
heading (3). Such functions we shall not call predicative, since predicative 
functions of individuals have already been defined as being such as are of the 
first order. Thus the order of a function is not determined by the order of its 
argument or arguments; indeed, the function may be of any order superior to 
the order or orders of its arguments. 

A variable matrix whose argument is <f> I z will be denoted by fl<f>lz, and 
generally, a matrix whose arguments are <f> I z, yjr I z, ... x, y, ... (where there is 
at least one function among the arguments) will be denoted by 

f\(4>\%^\%...SD,y,...). 
Such a matrix is not of the first or second order, since it contains the new 
variable /, whose values are second-order matrices. We proceed to construct 
new matrices as we did with the matrix <f> ! ot ; these constitute third-order 
matrices. These together with the functions derived from them by generali- 
zation are called third-order functions, and the propositions derived from third- 
order matrices by generalization are called third-order propositions. 

In this way we can proceed indefinitely to matrices, functions and propo- 
sitions of higher and higher orders. We introduce the following definition: 

A function is said to be predicative when it is a matrix. It will be 
observed that, in a hierarchy in which all the variables are individuals or 
matrices, a matrix is the same thing as an elementary function (cf. pp. 
127, 128). 

"Matrix" or "predicative function" is a primitive idea. 

The fact that a function is predicative is indicated, as above, by a note of 
exclamation after the functional letter. 

The variables occurring in the present work, from this point onwards, will 
all be either individuals or matrices of some order in the above hierarchy. 
Propositions, which have occurred hitherto as variables, will no longer do so 



SECTION B] THE AXIOM OF REDUCIBILITY 165 

except in a few isolated cases of which no subsequent use is made. In practice, 
for the reasons explained on p. 162, a function of a matrix may be regarded 
as capable of any argument which is a function of the same order and takes 
arguments of the same type. 

In practice, we never need to know the absolute types of our variables, but 
only their relative types. That is to say, if we prove any proposition on the 
assumption that one of our variables is an individual, and another is a function 
of order n, the proof will still hold if, in place of an individual, we take a 
function of order m, and in place of our function of order n we take a function 
of order n + m, with corresponding changes for any other variables that may 
be involved. This results from the assumption that our primitive propositions 
are to apply to variables of any order. 

We shall use small Latin letters (other than p, q, r, s) for variables of the 
lowest type concerned in any context. For functions, we shall use the letters 
<f>, ^ X> @>f> 9> ^(except that, at a later stage, F will be defined as a constant 
relation, and 6 will be defined as the order-type of the continuum). 

We shall explain later a different hierarchy, that of classes and relations, 
which is derived from the functional hierarchy explained above, but is more 
convenient in practice. 

When any predicative function, say <£ ! z , occurs as apparent variable, it 
would be strictly more correct to indicate the fact by placing " (<£ ! z) " before 
what follows, as thus: " (<f> I z) . f (<f> 1 z).' 1 But for the sake of brevity we 
write simply " (<£) " instead of "(<£ ! £)." Since what follows the <f> in brackets 
must always contain <f> with arguments supplied, no confusion can result from 
this practice. 

It should be observed that, in virtue of the manner in which our hierarchy 
of functions was generated, non-predicative functions always result from such 
as are predicative by means of generalization. Hence it is unnecessary to 
introduce a special notation for non-predicative functions of a given order and 
taking arguments of a given order. For example, second-order functions of an 
individual x are always derived by generalization from a matrix 

fl((f>lz, yfrlz, ... x, y, z, ...), 
where the functions/ <£, 1/r, . . . are predicative. It is possible, therefore, without 
loss of generality, to use no apparent variables except such as are predicative. 

We require, however, a means of symbolizing a function whose order is not 
assigned. We sjiall use "<f>x" or "/(% ! z)" or etc. to express a function (<f> or/) 
whose order, relatively to its argument, is not given. Such a function cannot 
be made into an apparent variable, unless we suppose its order previously fixed. 
As the only purpose of the notation is to avoid the necessity of fixing the order, 
such a function will not be used as an apparent variable; the only functions 
which will be so used will be predicative functions, because, as we have just 
seen, this restriction involves no loss of generality. 



166 MATHEMATICAL LOGIC [PART I 

We have now to state and explain the axiom of reducibility. 

It is important to observe that, since there are various types of propositions 
and functions, and since generalization can only be applied within some one 
type (or, by means of systematic ambiguity, within some well-defined and 
completed set of types), all phrases referring to "all propositions" or "all 
functions," or to "some (undetermined) proposition " or " some (undetermined) 
function," are prima facie meaningless, though in certain cases they are capable 
of an unobjectionable interpretation. Contradictions arise from the use of 
such phrases in cases where no innocent meaning can be found. 

If mathematics is to be possible, it is absolutely necessary (as explained 
in the Introduction, Chapter II) that we should have some method of making 
statements which will usually be equivalent to what we have in mind when 
we (inaccurately) speak of "all properties of x." (A "property of x" may be 
defined as a prepositional function satisfied by x.) Hence we must find, if 
possible, some method of reducing the order of a propositional function without 
affecting the truth or falsehood of its values. This seems to be what common- 
sense effects by the admission of classes. Given any propositional function -frx, 
of whatever order, this is assumed to be equivalent, for all values of x, to a 
statement of the form "x belongs to the class a." Now assuming that there 
is such an entity as the class a, this statement is of the first order, since it 
involves no allusion to a variable function. Indeed its only practical advantage 
over the original statement yfrx is that it is of the first order. There is no 
advantage in assuming that there really are such things as classes, and the 
contradiction about the classes which are not members of themselves shows 
that, if there are classes, they must be something radically different from in- 
dividuals. It would seem that the sole purpose which classes serve, and one 
main reason which makes them linguistically convenient, is that they provide 
a method of reducing the order of a propositional function. We shall, therefore, 
not assume anything of what may seem to be involved in the common-sense 
admission of classes, except this, that every propositional function is equivalent, 
for all its values, to some predicative function of the same argument or argu- 
ments. 

This assumption with regard to functions is to be made whatever may be 
the type of their arguments. Let fu be a function, of any order, of an argument 
u, which may itself be either an individual or a function of any order. If / is 
a matrix, we write the function in the form flu; in such a case we call / a 
predicative function. Thus a predicative function of an individual is a first- 
order function; and for higher types of arguments, predicative functions take 
the place that first-order functions take in respect of individuals. We assume, 
then, that every function of one variable .is equivalent, for all its values, to 
some predicative function of the same argument. This assumption seems to 
be the essence of the usual assumption of classes; at any rate, it retains as muoh 



SECTION B] THE AXIOM OF REDUCIBILITY 167 

of classes as we have any use for, and little enough to avoid the contradictions 
which a less grudging admission of classes is apt to entail. We will call this 
assumption the axiom of classes, or the axiom of reductibility. 

We shall assume similarly that every function of two variables is equivalent, 
for all its values, to a predicative function of those variables, i.e. to a matrix. 
This assumption is what seems to be meant by saying that any statement about 
two variables defines a relation between them. We will call this assumption 
the axiom of relations or (like the previous axiom) the axiom of reducibility. 

In dealing with relations between more than two terms, similar assumptions 
would be needed for three, four, . . . variables. But these assumptions are not 
indispensable for our purpose, and are therefore not made in this work. 

Stated in symbols, the two forms of the axiom of reducibility are as follows: 
*121. h: (a/) :^t. =,./!* Pp 

*1211. h:(zf):<l>(x,y).= x>y .fl(x,y) Pp 

We call two functions <f>%, ty& formally equivalent when <f>x.^. x . yfrx, and 
similarly we call <£ (oc, §) and yfr {x, §) formally equivalent when 

<f>(x } y).= x>y .yjr(x ) y). 
Thus the above axioms state that any function of one or two variables is 
formally equivalent to some predicative function of one or two variables, as 
the case may be. 

Of the above two axioms, the first is chiefly needed in the theory of classes 
(*20), and the second in the theory of relations (#21). But the first is also 
essential to the theory of identity, if identity is to be defined (as we have done, 
in #13'01); its use in the theory of identity is embodied in the proof of #13101, 
below. 

We may sum up what has been said in the present number as follows: 

(1) A function of the first order is one which involves no. variables except 
individuals, whether as apparent variables or as arguments. 

(2) A function of the (n + l)th order is one which has at least one argument 
or apparent variable of order n, and contains no argument or apparent variable 
which is not either an individual or a first-order function or a second-order 
function or ... or a function of order n. 

(3) A predicative function is one which contains no apparent variables, 
i.e. is a matrix. It is possible, without loss of generality, to use no variables 
except matrices and individuals, so long as variable propositions are not 
required. 

(4) Any function of one argument or of two is formally equivalent to a 
predicative function of the same argument or arguments. 



*13. IDENTITY 

Summary q/"#13. 

The prepositional function "x is identical with y" will be written "x — y." 
We shall find that this use of the sign of equality covers all the common uses 
of equality that occur in mathematics. The definition is as follows: 

♦1301. #=y. = :(<£):0!#.D.<£!> Df 

This definition states that x and y are to be called identical when every 
predicative function satisfied by x is also satisfied by y. We cannot state that 
every function satisfied by # is to be satisfied by y, because x satisfies functions 
of various orders, and these cannot all be covered by one apparent variable. 
But in virtue of the axiom of reducibility it follows that, ifx = y and x satisfies 
tyx, where yjr is any function, predicative or non-predicative, then y also satisfies 
i]ry (cf. ♦13101, below). Hence in effect the definition is as powerful as it 
would be if it could be extended to cover all functions of x. 

Note that the second sign of equality in the above definition is combined 
with "Df," and thus is not really the same symbol as the sign of equality 
which is defined. Thus the definition is not circular, although at first sight 
it appears so. 

The propositions of the present number are constantly referred to. Most 
of them are self-evident, and the proofs offer no difficulty. The most important 
of the propositions of this number are the following: 

♦13101. \-\x = y."^.^rxZ>^y 

I.e. if x and y are identical, any property of x is a property of y. 

♦1312. )r:x^y.D.ylrx = yfry 

This includes ♦13101 together with the fact that if x and y are identical 
any property of y is a property of x. 

♦13151617, which state that identity is reflexive, symmetrical and transitive. 

♦13191. h :.y = x.D y .(f>y : = .<f>x 

I.e. to state that everything that is identical with x has a certain property 
is equivalent to stating that x has that property. 

♦13195. h : (gy) .y = x.<f>y . = .<f>x 

I.e. to state that something identical with x has a certain property is 
equivalent to saying that x has that property. 

♦13*22. h : (32, w).z — x.w = y.<li(z,w).= .<f>(x,y) 
This is the analogue of ♦13*195 for two variables. 



SECTION b] identity 169 

*1301. x = y. = :(<f>):<f>lx.'2.<f>ly Df 

The following definitions embody abbreviations which are often convenient. 
*1302. x^y. = .~(x = y) Df 

#1303. x = y = z. = .x = y.y = z T)f 

*131. \-:.a} = y. = :<f>l£c.3t.<l>ly [*4'2 . (*13-01) . (*10'02)] 
*13'101. h : x = y . D . yjrx D <^y 

Bern. 
V . #121 . D h :. (a<£) :. yfrx . = .</>! # : ■fy . = . <f> I y (1) 

J- . *131 . D r :: Hp . D :. <f> ! x . D* . <£ ! y :. 

[*4'84-85.*10-27] D :. tyx . = . cf> ! x : ^y . = . <f> ! y : D* : ^a; . 3 . i/ry :. 
[*10-23] D:.('g i (f)):-^x. = .<f>lxiylry. = .<f>ly:D:yfrx.D.'s{ry (2) 

H.(l).(2).Dh.Pro.p 

In virtue of this proposition, if x = y, y satisfies any function, whether 
predicative or non-predicative, which is satisfied by x. It will be observed 
that the proof uses the axiom of reducibility (#12'1). But for this axiom, two 
terms x and y might agree in respect of all predicative functions, but not in 
respect of all non-predicative functions. We should thus be led to identities 
of different degrees, according to the degree of the functions in respect of 
which x and y agreed. Strict identity would, in this case, have to be taken as 
a primitive idea, and #13101 would have to be a primitive proposition, as would 
also *13151617. 

#1311. \--.:x = y. = i<l>lx.=4 > .<l>ly 
Bern. 

h . #1022 . 3\-:.<f>lx.=4 > .<f>ly:0:<f>lx.1 < i,.<l>ly: 

[*13-1] D:x = y (1) 

K #13101. 3\-:.x = y.3.<l>lx3<f>ly (2) 

I- . *13'101 . #1-7 . D\-:.x = y.D.~<j>lx1~<f>ly. 

[Transp] D .<j>lyD<j>lx (3) 

h.(2).(3).Comp.Dh:a; = y. D .<f>lx=<j>ly: 

[*10'ir21] DH:.# = 2/*D:<£!#.=*.0!y (4) 

K (1) . (4) . D h . Prop 

$1312. H : x = y . D . <*}rx = ijry 
Bern. 

Y . #13'101 . Comp .Dha! = jf.D. tyx D ifry . ~yfrx D r^yfry . 





[Transp] 


D . yfrx = yjry : D H . Prop 


#1313. 


V : yjrx . x = y .O . yfry 


[#13101 . Comm. Imp] 


*1314. 


b : yfrx. f^yfry . D . x^y 


[#13-13. #414] 


#1315. 


)r ,x = x 


[Id. #1011. #131] 


#1316. 


Y \x — y. =.y — x 


[#1311. #1032] 



170 MATHEMATICAL LOGIC [PART I 

#1317. Y:x = y.y = z.D.x — z 

Dem. 

V . #131 . D h :: Hp . D :. <f> ! x . D* . <j> ! y : ! y . D$ . <f> ! z.\. 
[*103] D :. <£ ! # . D^ . ! * :: D h . Prop 

In the above use of #103, <f> ! x, <£ ! y, <j> ! z are regarded as three different 
functions of <£, and <f> replaces the x of #10-3. 

The above three propositions show that identity is reflexive (#1315), 
symmetrical (#1316), and transitive (#13-17). These are the three marks of 
relations having the formal properties which we associate commonly with the 
sign of equality. 

*13171. h:x = y.x = z.D.y = z [*13-16\l7] 

*13172. Y:y = x.z = x.0.y = z [*131617] 
#1318. h:x = y.x^z.D.y^z [#13-17 .#414] 
*13181. h:x = y.y^z.D.x^z [#13-171 . #414] 

#13182. \-:.x = y.D:z = x. = .z = y [#1317172 . Exp. Comp] 
#13183. h :. x = y . = ; z = x . = z . z ~ y 

Dem. 

h. #13182. #1011-21.3 1- :.#=?/. Z> :z = x.= z .z = y (1) 

h . #10'1 . Dh:. z = x.=g.z = y:D: x = x.D.x = y: 
[*13*15] 3:x = y (2) 

h . (1) . (2) . D K Prop 

#1319. r-.(ay).y = a> [*13'15 . #10-24] 

#13-191. h:.y = x.D y .(f>y: = .<j)x 

Dem. 

r- . #10'1 . D b :. y = x . D y . tf>y : D : x = x i D . <f>x : 

[#13-15] D : <f>x (1) 

h.*1312. Db:.y = x.D:<f)x. ^.<f>y:. 
[Comm] D h :. <f>x . D : y = x. D.<f>y:. 

[#10-11-21] D\-:.<j>x.D:y = x. D y .<f>y (2) 

h . (1) . (2) . D H . Prop 
This proposition is constantly used in subsequent proofs. 
#13-192. h :. (gc) : x = b . = x . x = c : sjrc : = . yjrb 
Dem. 

f- . #4-2 . #3-2 .D\-::fb.D:.x=b.= x .x = b:^b:. 
[#1024] " D:.(^c):x=b.= x .x = c:yfrc (1) 

f- .#101 .0\- :.x = b .= x .x — c zyjrc :"D :b = b . = ,b = c:\lrc: 
[*5-501.*13-15] D:6 = c.^c: 

[#13-13] D:^/r& (2) 

h . (2) . #10-11-23 . D h :. (gc) : x= b . = x . x = c : ^c : D . yfrb (3) 

K(l).(3).Dh.Prop 
This proposition is useful in the theory of descriptions (#14). 



SECTION B] 



IDENTITY 



171 



#13193. h : <f>x . x \ = y . = . <f>y . x = y 
Dem. 

h . Simp . Dh:(f}x.x = y.D.x = y 

K #13*13. 3\-:<f>x.x = y.1.<f>y 

h . (1) . (2) . Comp .D\- m .<j>x.x = y.'D.<l>y.x = y 

h . #1316 . Fact . Dh : (j>y .x = y .D . <f>y .y = x . 

(3) |^1. D.<f>x.y = x. 

*1316.Fact] D.<f>x.x = y 

h . (3) . (4) . D h . Prop 

This proposition is very often used. 

#13194. h :(f>x. x - y . = .<f>x. (f>y .x = y [#13*13 . #4*71] 

This proposition is used in #37*65 and #101*14. 

#13*195. b : (ay) .y = x . <f>y . = ,<f>x 

Dem. 

h . #3*2 . #13*15 . D\-:<f>x.D.x = x.<j)X. 

[#10*24] D.(ny).y = x.<f>y 

\- . #1313 . #1011 . D h :. (y) : y = as . <\>y . D . <f>x : 

[#10*23] D H :. (ay) . r/ = * . cf>y . D . <\>x 

h . (1) . (2) . D h . Prop 

The use of this proposition in subsequent proofs is very frequent. 

#13*196. h :. ~(f>x . = i(f>y. , Dy.y^x [#13*195 . Transp . #10*51] 
#13*21. h :.z = x .w = y .D ZiW .<j>(z,w): = .<f>(x,y) 
Dem. 
h. #11*62.3 



(1) 
(2) 
(3) 



(4) 



(1) 

(2) 



. z = x . D z : w = y . O w . <\> (z, w) :. 
.w = y.D w .<f>(x,tv):. 
. <f>(x,y)::D\- .Prop 



h :: z = x . w = y . D ZjW . <£ (z, w) : = 
[#13*191] = 

[#13*191] = 

This proposition is the analogue, for two variables, of #13*191. 
#13*22. h : (a^, w) . z = x . w = y . <£ (z, w) . = . (f> (x, y) 
Dem. 
h . #11 *55 . D h : . (a^, w) . z = x . w = y . <f> (z, w) . 

(a-^) : z = x : (aw) .w = y .<f>(z, iv) : 
[#13*195] 
[#13*195] 



(aw) .iv = y.<f>(x,w): 
4>(x, y) :. D I- . Prop 



This proposition is the analogue, for two variables, of #13195. It is fre- 
quently used, especially in the theory of couples (#54, #55, #56). 

The following proposition is useful in the theory of types. Its purpose is 
to show that, if a is any argument for which " <f>a " is significant, i.e. for which 
we have <f>avo*><f>a, then "<f>x" is significant when, and only when, x is either 



172 MATHEMATICAL LOGIC [PART I 

identical with a or not identical with a. It follows (as will be proved in #20*81) 
that, if "<j>a" and "tya" are both significant, the class of values of # for which 
"<f>x" is significant is the same as the class of those for which "yfrx" is signi- 
ficant, i.e. two types which have a common member are identical. 

In the following proof, the chief point to observe is the use of #10*221. 
There are two variables, a and x, to be identified. In the first use, we depend 
upon the fact that. $a and x = a both occur in both (4) and (5) : the occurrence 
of <f>a in both justifies the identification of the two a's, and when these have 
been identified, the occurrence of x = a in both justifies the identification of 
the two x's. (Unless the a's had been already identified, this would not be 
legitimate, because "x = a" is typically ambiguous if neither x nor a is of 
given type.) The second use of #10*221 is justified by the fact that both <f>a 
and (f>x occur in both (2) and (6). 
#13*3. h ::<f>av <**><f>a .D :.<f>xv ™<f>x . = :x — a.v .x^a 

Dem. 

h.*2*ll. Dh.^v-v^B (1) 

h.(l).Simp. D I- : <£av~<£a. D .<j>xv~<f)x (2) 

f-.*211. Dhzx = a.v.x^a (3) 

H. (3). Simp. D I- :. </»av~^a, D :x = a. v.x^a (4) 

r-. #13*101 . Comm . Dh :. <£av~0a . D :x = a . D . <f>xv<^><f>x (5) 

I- . (4) . (5) . *1013*221 . D 

J*::^av~^a.D:a; = a.v.a!^a:. ^av<vfi . D : x = a. D . <j>xv ™$x (6) 

h . (2) . (6) . #1013*221*. D 

h :: ^av~^a.D . (f>xv™<f>x:.<l>av<^<f>a . D :# = a. v ,x=\=a:. 

d>avr^d>a.^>:x = a.D.6xvr^(bx (7) 
K (7). Simp.} 

I-:: <j>av oj(jxi . D . <f>x v~<jkc :.^v~^a. D : # = a . v . a?4=a (8) 

t- i (8). #6*35 . D f- :: <f>av<^^>a . D :. <f)xv™<fix . -=:# = ». v. #={= a:: 

D h . Prop 



*14. DESCRIPTIONS 

Summary of #14. 

A description is a phrase- of the form " the term which etc.," or, more 
explicitly, " the term x which satisfies <f>&," where <£& is some function satisfied 
by one and only one argument. For reasons explained in the Introduction 
(Chapter III), we do not define " the x which satisfies <f>$," but we define any 
proposition in which this phrase occurs. Thus when, we say : " The term x 
which satisfies <f>x satisfies yjrx," we shall mean : " There is a term b such that 
<f>x is true when, and only when, x is b, and tyb is true." That is, writing 
" (ix) (cf>x) " for " the term x which satisfies <f>x," yjr (ix) (<f>x) is to mean 

(a&) : <f>x . = x . x = b : yfrb. 

This, however, is not yet quite adequate as a definition, for when (ix) (<f>x) 
occurs in a proposition which is 'part of a larger proposition, there is doubt 
whether the smaller or the larger proposition is to be taken as the "^(ix)(<f>x).' K 
Take, for example, ty(ix) (<f>x) . D . p. This may be either 

(a&) : <px . = x . x == b : yjrb : D .p 
or {'&b):.<f>x.= x .x = b:'fb.Z).p. 

If " (a&) : <f>x . = x . x = b " is false, the first of these must be true, while the 
second must be false. Thus it is very necessary to distinguish them. 

The proposition which is to be treated as the " yjr (ix) (<f>x) " will be called 
the scope of (ix)(<f>x). Thus in the first of the above two propositions, the 
scope of (ix) (<f>x) is yfr (ix) (<f>x), while in the second it is yjr (ix) (<£#) . D . p. 
In order to avoid ambiguities as to seope, we shall indicate the scope by 
writing " [(w)(<f>x)]" at the beginning of the scope, followed by enough "dots 
to extend to the end of the scope. Thus of the above two propositions the 
first is 

[(ix) (<f>x)] . f (ix) (<j>x).D.p, 
while the second is 

[(ix) (<f>x)] : i/r (ix) (<f>x) .D.p. 

Thus we arrive at the following definition : 

*1401. [(ix)(<\>x^\.^(ix)(4>x) . = :(a&):<^ .= x .x = b:yfrb Df 

It will be found in practice that the scope usually required is the smallest 
proposition enclosed in dots or brackets in which " (ix) (<f>x) " occurs. Hence 
when this scope is to be given to (ix) ($x), we shall usually omit explicit 
mention of the scope. Thus e.g. we shall have 

aJr(ix)(<f>x) . = : ( a &) : fa .= x . x = b : a^b, 
~ [a = (ix) (<f)x)} . = . ~ {(g6) : $sc . = x . x = b : a = b). 



174 MATHEMATICAL LOGIC [PART I 

Of these the first necessarily implies fab) : <f>x .= x .x = b, while the second 
does not. We put 

*1402. El(ix)(<f>x).^:fab):<f>x.= x .x = b Df 

This defines : " The x satisfying <f>£ exists," which holds when, and only 
when, <f>ai is satisfied by one value of x and by no other value. 

When two or more descriptions occur in the same proposition, there is 
need of avoiding ambiguity as to which has the larger scope. For this purpose, 
we put 

*1403. [(ix) (<j>x), (ix) (yfrx)] ./{(ix) (<f>x), (ix) (yjrx)} . = : 

[(ix)(<f>x)] : [(ix) (fx)] ./{(ix) (<j>x), (ix) (fx)} Df 

It will be shown (#141 13) that the truth- value of a proposition containing 
two descriptions is unaffected by the question which has the larger scope. 
Hence we shall in general adopt the convention that the description occurring 
first typographically is to have the larger scope, unless the contrary is expressly 
indicated. Thus e.g. 

(lx)(<f>x) = (lx)(yfrx) 

will mean fab) : <f>x .= x .x = b:b = (ix) (yfrx), 

i.e. fab) :. <\>x . = x . x = b :. fac) : tyx .= x .x = c:b = c. 

By this convention we are able almost always to avoid explicit indication of 
the order of elimination of two or more descriptions. If, however, we require 
a larger scope for the later description, we put 

*1404. [(ix) (fx)] ./{(ix) (cf>x), (ix) (+x)} . = . 

[(ix)(^x) } (ix) (<f>x)] ./{(ix) (#;), (i*) drx)} Df 

Whenever we have E ! (ix)((f>x), (ix)($x) behaves, formally, like an ordinary 
argument to any function in which it may occur. This fact is embodied in 
the following proposition : 

*1418. h : . E ! (ix) (<f>x) . D : (x) . fx . D . f (ix) (<j>x) 

That is to say, when (ix) (<f>x) exists, it has any property which belongs to 
everything. This does not hold when (ix) (<f>x) does not exist ; for example, 
the present King of France does not have the property of being either bald 
or not bald. 

If (ix) (<f>x) has any property whatever, it must exist. This fact is stated 
in the proposition : 

*14'21. h:^(ix)(<f>x).D.El(ix)((f>x) 

This proposition is obvious, since " E ! (ix) (<px) " is, by the definitions, part 
of " yjr (ix) (4>x)." When, in ordinary language or in philosophy, something is 
said to "exist," it is always something described, i.e. it is not something 
immediately presented, like a taste or a patch of colour, but something like 
" matter " or " mind " or " Homer " (meaning " the author of the Homeric 



SECTION b] descriptions 175 

poems "), which is known by description as " the so-and-so," and is thus of 
the form (ix) (<f>x). Thus in all such cases, the existence of the (grammatical) 
subject (ix) (<f>x) can be analytically inferred from any true proposition having 
this grammatical subject. It would seem that the word " existence " cannot 
be significantly applied to subjects immediately given ; i.e. not only does our 
definition give no meaning to " E ! x," but there is no reason, in philosophy, to 
suppose that a meaning of existence could be found which would be applicable 
to immediately given subjects. 

Besides the above, the following are among the more useful propositions 
of the present number. 

*14 202. h :. <f>oc . = x . x = b : = : (ix) (cf>x) = b : = : (f>x . = x . b = x : = : b — (ix) (<f>x) 

From the first equivalence in the above, it follows that 
*14-204. r : E ! (ix) (<j>x). = . ( a 6) . (ix) (<f>x) = b 

I.e. (ix) (<f>x) exists when there is something which (ix) (<f>x) is. 

We have 
*14 205. r- : yjr (ix) (<j>x) . = . (36) . b = (ix) (<f>x) . tyb 

I.e. (ix) (<f)x) has the property yfr when there is something which is (ix) (<f>x) 
and which has the property yjr. 

We have to prove that such symbols as " (ix) (<j>x) " obey the same rules 
with regard to identity as symbols which directly represent objects. To this, 
however, there is one partial exception, for instead of having 

(ix) ((f>x) = (ix) (<ja), 
we only have 

*14-28. r- : E ! (ix) ($x) . = . (ix) (<f>x) = (ix) ($x) 

I.e. " (ix) (4>x) " only satisfies the reflexive property of identity if (ix) (<f>x) 
exists. 

The symmetrical property of identity holds for such symbols as (7#) (<f>x), 
without the need of assuming existence, i.e. we have 
#1413. h : a = (ix) (<j>x) . = . (ix) (<px) = a 
*14'131. I" : (ix) (<f>x) = (ix) (tyx) . = . (ix) (yjrx) = (ix) (<f>x) 

Similarly the transitive property of identity holds without the need of 
assuming existence. This is proved in #1414'142144. 



*14-01. [(ix)(<f>x)].f(ix)(<l>x). = :(Rb):<l>x.= x .x = b:fb Df 
*1402. E\(ix)((f>x). = :('g i b):<j>x.= x .x = b Df 

*14-03. [(ix) (4>x), (ix) tfx)] .f{(ix) (cf>x), (ix) (yjrx)} . = : 

[(ix) (4>x)] : [(ix) (W] ./{(ix) ($x), (ix) (yjrx)} Df 
*14"04. [(ix) (fx)] ./{(las) (<f)x), (ix) (yjrx)} . = . 

[(lx)(y\rx), (ix)(<f>x)] ./{(ix)(cj>x), (lx)(yfrx)} Df 



176 



MATHEMATICAL LOGIC 



[PART I 



#141. b :. [{ix) {<f>x)] . yjr (ix) (<f>x) . = : (36) : <f>x . = x . x = b : yjrb 

[*4-2.(*1401)] 
In virtue of our conventions as to the scope intended when no scope is 
explicitly indicated, the above proposition is the same as the following : 

#14101. I- :. -f (ix) (<f>x) . = : (36) : (f>x . = x . x = b : tyb [#141] 
#1411. H :. E ! (ix) (<f>x) . = : (36) :<f>x.= x .x = b [*4-2 . (*14'02)] 
#14111. V :. i(w) (yfrx)] ./{(ix) (<f>x), {ix) (fx)} . = : 

(36, c) : <f>x . = x . x = b : yfrx . = x . x = c :/(&, c) 
Dem. 
h . #4-2 . (*14'04-03) . D 
h :: [(ix) (+x)] ./{(?*) ($*), (1*) (**)} . = :. 

{{ix) (yfrx)] : [{ix) (**)] ./{(**) (**), (w) (^)j :. 
[#141] = :. [(M)ty*)] » (a 6 ) : ** • =* ^ = b:f{b, (ix)(^x)} :. 
[#141] = :. (3c) :.tyx.= x .x = ci. (36) : <£# . = x .x = b:f{b, c) :. 
[#11-55] = :. (36, c) : <j>x .= x .x = c : yjrx . = x .x = b :/(&, c) :: D h . Prop 

#14112. H :./{(?«>) (<H> (w)(^)} ■ = : 

(36, c) : <£# . = x . x = b : yfrx . = x . x = c :/{b, c) 
[Proof as in #14111] 
In the above proposition, we assume the convention explained on p. 174, 
after the statement of #14 - 03. 

#14113. h : [{ix) (fx)] ./{{ix) (<f>x), (ix) (fx)} . = ./{(«) (<f>x), (ix) (f *)} 
[#14111112] 

This proposition shows that when two descriptions occur in the same pro- 
position, the truth- value of the proposition is unaffected by the question which 
has the larger scope. 
#1412. H :. E ! {ix) {<f>x) . D : <f>x .<f>y . D x>y .x = y 

Dem. 
K*1411 Dl-:.Hp.D:(a6):^aJ.s«.a? = 6 (1) 

K #438. #101. #1111-3. D 
V :. (}>x . = x .x = b : D : <$>x . $y . = x>y .x = b .y=b. 

[#13172] D x , y .x = y (2) 

h . (2) . #10-11-23 . 3 h :. (36) : <f>x . s. . x = b : D : $x . <f>y . D XtV . x = y (3) 
K(l).(3). Dh.Prop 

#14-121. h :. 4>x . = x . x = b : $0 . = a . oc- c : D . b = c 

Bern. 

h . #101 .Dhz.Hp.D: <£&. = .& = &:#>. = .6 = c: 

[#1315] Di<f>b:<f>b. = .b = o: 

[Ass] D : 6 = c:.DH.Prop 

#14*122. \- :. (frx . = x . x=b : = : <j>x .O x . x = b : <f>b : 

= :<f>x .D x .x = b: (30) . <£# 



SECTION b] descriptions 177 

Bern. 

V . #10'22 . D h :. <j>x . = x . x = b : = : <f>x . "2 X . x = b : x = b . D«, . <f>x : 
[*13'191] = : <j>x . Z>* . x = b : <£& (1) 

K*4-7l. DK :.<£#. D .x-b: D:<j>x.= .<f>x, x = b:. 

[*1011-27] 2 \- ;. <]>x .2 X . x = b :D : <f>x . = x . <f>x . x = b : 
[*1 0-281] D : fax) .<f>x. = . fax) . <f>x . x = b . 

[#13195] - = .</>& (2) 

h . (2) .#532 . 3 h :. <f>x . D x . x = b : fax) .<f>x: = : <f>x . D x . x = b : <j>b (3) 

K(l).(3). DKProp 

The two following propositions (#14123-124) are placed here because of 
the analogy with #14122, but they are not used until we come to the theory 
of couples (#55 and #56). 

*14'123. H :. <£ (z, w) . = z >w . z = x . w = y : 

= :<f>(z,w). D z>w .z = x.w = y:<f>(x,y): 

= :<f>(z,w). D ZiW .z = x.w = y: faz, w) . <j> (z, w) 
Bern. 

Y .#11-31 . D Y :.<f>(z, w) .= ZyW . z = x .w = y : 

= :<f>(z,w). ZiW .z = x.w = y:z = x.w = y. D ZtW . $ (z, w) : 

[*13-21] = : <f>(z, w) . X,w • z = x . w = y : <f>(x, y) (1) 

K*4-71. 3\-:.<f>(z,w).'5.z = x.w = y: 

D : <f> {z, w) . = . <f> (z, w) . z = x . w — y :. 
[#1111-32] D h :. <£ (z, w) . D ZtW .z = x.w = y: 

D:<f>(z, w).= ZtW .<f>(z,w).z = x.w = y: 

[#11-341] D : faz, tv) . <f> (z, w). = . faz, iv) .<f>(z, w) .z = x .w = y . 

013-22] =.<f>(x,y) (2) 

I- . (2) . #5-32 . D h :. (z, w) . 5 Z>W .z = x.tv = y :faz, w) .<f>(z, w) : 

I- n\ /ox ^ L x> ='<Kz,w).1z,v,.z = x.w = y.<l>{x } y) (3) 

r .(l).{6) . Jb. Prop 

#14124. h :. fax, y) \<\>{z, w) . = ZyW .z = x.w = y: 

= '• (a*» y) -4>(x> y) -<f>(^, w) .<f>(u, v) . D ZtW>u>v .z= u .w=v 

Dem. 
V. #14123. #327.D 

•" '•{'Spc, y) : (f> (z, w) . = ZjW .z = x.w = y:D. fax, y) . <f> (x, y) (1) 
h . #11-1 . #3-47 . D h :. $ (z, w) . = z>w .z = x.w = y: 

D : <p (z, tv) . <f> (u, v) . D . z = x . w = y . u = x . v = y . 
[#13-172] D.z = u.w = v (2) 

K (2). #1111-35. D 
r- :. fax, y) : <j> (z, w) . = ZtW .z = x.w = y: 

D:<f>(z ) w).<f)(u,v).D ,z = u.w = v (3) 



R&W I 



12 



178 MATHEMATICAL LOGIC [PART I 

h.(3).*ll'll-3.D 

\-'..(^x,y)i^(z i w).= ZtW .z = x.w = y: 

D : <j> (z, w) . <f> (u, v) . O z , w ,u,v .z = u.w = v (4) 
h . #11-1 . 3 H :. <f> (x, y) : <j> (z, w) . <j> (it, v) . z>Wl u, v .z = u.w = v: 

0:<f>(x, y) : <f> (z, w) . <£ (x, y) . D Zt w . z = x . w = y : 
[*5-33] D : (f> (x, y) : <j> (z, w) . D z> w . z = x . w = y: 

[*1 4-123] D-.(j>(z > w).= z>w ..z = x.w = y (5) 

h.(5).*iril'34-45.D 

I" =■ (a#> y) >4>(x,y):4> 0, w) . <f> (u, v) . O z , w> u,v .z = u.w = v: 

D:(Rx,y):<l>(z,w).= z>w .z = x.w = y (6) 
h . (1) . (4) . (6) . D K Prop 

*1413. h : a = (ix) (<j>x) . = . (ix) (<f>x) = a 

Dem. 
h.*14-l. D\-:.a-=(ix)(<f)x). = :('3b):<f)X.= x .x = b:a = b (1) 

h . *1316 . *436 .1 \- :. <f>x . = x . x=b :a = b : = : <f>x .= x . x=b :b = a : 
[*10'11-281] D I- :. (a&) :(f>x.= x .x = b:a = b: 

— '■ i'Sfi) '• <f>x * =z • ® — b : b = a : 
[*14-1] = :(ix)(<fix) = a (2) 

h . (1) . (2) . D h . Prop 

This proposition is not an immediate consequence of #13 16, because 
"a = (ix)(<j>x)" is not a value of the function "x = y." Similar remarks 
apply to the following propositions. 
*14131. r : (ix) (<j>x) = (ix) (yfrx) . = . (ix) (yfrx) = (ix) (<f>x) 

Dem. 
\- . *14-1 . D h :: (ix) (<j)x) = (ix) (yfrx) . = :. (a&) : <f>x . =^ . x = b : b = (ix) (fx) :. 
[*14'1] = :. (g» :. 4>x .= x .x = b :. (gc) :^x.= x . x=c : 6 = c :. 
[*H-6] = :. (ac) :. -fx . = x . x = c :. (g6) :<f>x .= x .x = b:b = c :. 
[*141] = :. (gc) :.yfrx.= x .x = c: (ix) (<f>x) = c :. 
[*1413] = :. (gc) :. -fx .= x .x = c : c = (?#) (</>«) :. 
[*14'1] = :. (?#) (^#) = (ix) (<f>x) :: D h . Prop 

In the above proposition, in accordance with our convention, the descriptive 
expression (ix) ($x) is eliminated before (ix) (yfrx), because it occurs first in 
"(ix)(<f>x) = (ix)(yjrxy'; but in " (ix) tyx) = (ix) ((f>x )," (ix)(-*jrx) is to be first 
eliminated. The order of elimination makes no difference to the truth-value, 
as was proved in #14113. 

The above proposition may also be proved as follows: 
H . #141 11 . D t- i. (ix) (cf>x) = (ix) (yfrx) . 

- : (3^' c): <f>x . = x .x = b: yfrx .= x .x=cib = c: 
[#4-3.*13-16.*l 11 1341] = :(Qb,c):fx.= x .x = c: <f>x . = x .x = b : c = b : 

[*11-2.*14111] = : (ix) (yfrx) = (ix) (<^») :. D V . Prop 



SECTION b] descriptions 179 

#1414. h : a = b . b = (ix) (<f>x) .D.a = (ix) (<f>x) [*1313] 
#14142. Via — (ix) (<f>x) . (ix) (<f>x) = (ix) (yjrx) .D.a = (ix) (sjtx) 
Bern. 

V . #141 . D H : : Hp . D : . (g&) :'<f>x.= x .x = b:a = b:. 

(gc) : <px . = x ~x = c : c = (ix) (\Jrx) :. 
[#13'195] D :.<f>x ,= x .x = a:. (gc) :<f>x.= x .x=czc = (ix) (tyx) :. 

[#10-35] D :. (gc) ;. <f>x . = x . x = a : <}>x . = x . x = c : c = (ix)(yfrx):. 

[#14121] D :.(gc) :.<£#. = x .a; = a. : a = c : c = (ix) (yjrx) : . 

[*3-27.*13195] D :. a = (ix) (yfrx) :: D H . Prop 

#14144. h : (ix) (<j>x) = (ix)(yfrx) . (ix) (yjrx) = (ix) (xx) . D . (ix) (<f>x) = (ix) ()(x) 

Dem. 
\- . #14111 . D I- :: Hp . D :. (ga, b) : <j)x ,= x .x = a: yfrx . = x .x = b :a = b :. 

(gc, d) : ^c . = x . x = c : x x • -x • x = d : c = d :. 
[#13195] D :. (ga) : 4>x . = x . x = a : yjrx. = x .x = a :. 

(gc) ityx ,= x . x= c \ x x •-*' x ~ c '" 
[#11-54] D :. (ga, c) : <f)X . = x . x = a : yfrx .= x ,x = a: 

yjrx .= x .x=c: ^x . = x .x = c :. 
[#14-121. #11*42] D :.(ga,e): <£# . = x .x = a : x# . = x .x — c : a = c :. 

[#14111] D:.(ix)(4>x) = (ix)(xx)::1\-.~Proy 

#14145. h : a = (•?#) (<f>x) . a = (?#) (yfrx) . D . (ix) (<f>x) = (ix) (yjrx) 
Dem. 

V . #141 . D I- :. a = (ix) (<f>x) . = : (g&) : <£# . =3 . x = 6 : a = 6 : 
[#13195]. =:<f>x.= x .x = a (1) 
H . (1) . #14-1 . D I- :: Hp . = :. ipx . = x . x = a :. (gi) : ^a; . -a, . x = 6 : a = 6 :. 
[#10-35] = :. (g&) :.<f>x.= x .x = a: -yjrx .= x .x = b:a = b:. 
[#14-111] D :. (?#) (<f>x) = (?#) (^) :: D I- . Prop 

#1415. r- :. (ix) (<j>x) = b.D:yfr {(ix) (<f>x)} . = :fb 

Dem. 

h.*141.0 

h::Hp. D :. (gc) : <j>x . = x .x = c:c = b :. 

[*13-195]D:.^.= x .a; = 6 (1) 

K (1) . #141 . D 

I- :: Hp . D:.yjr {(ix) (<f>x)} . = : (gc) :x = b.= x . x = c : y]rc : 

[#13-192] =:-^6::Dh.Prop 

#14-16. h :. (ix) (cf>x) = (ix) (tyx) . D : x {(ix) (</>#)} . = . x {(»«) (^)} 

Dem. 
\- . #14-1 . D h :. Hp . D : (g&) : (f>x . = x . x = b : b = (ix) (yjrx) (1) 

h . #14 - 1 . D h :: fyx . = x . x = b : D :. 

X (O) ( Wl • = : (a c ) : x = & ■ =* ■ * = c : %c : 
[#13-192] = : X l> (2) 

12—2 



180 MATHEMATICAL LOGIC [PART I 

r . *14\L315 . D r :.b = (w) (fa?) .3: x b. = . x ((**) (fx)} (3) 

K(2).(3). 0\-:.<f>x.= x .x = b:b = (ix)(yftx): 

D : X {(in) (<f>x)} . = . X {(*#) (^)l (4) 
H.(l).(4).*10-l-23.DKProp 

*1417. I" : . (ix) (0a?) = b . = : yfr I (ix) (0a?) .^.yjrlb 

Bern. 
K*1415.*10ir21.D 

H:.(?a?)(0a?)-&.D:^!(?a?)(0a?).=*.^!& (1) 

\- . *101 . *422 .Db::xlx.= x .x=b:ylrl (ix)(<f>x) . =* . -0- ! & : 

D:(?a?)(0a?) = 6. = .& = &: 
[*13'15] D:(?a?)(0a?) = 6 (2) 

h.(2).Exp.*1011'23.D 

b::(^ X ):x^-=x-^ = b:D:.yjrl(ix)(<l>x).= ri/ .ylrlb:^.(ix)( ( f i x) = b ( 3 ) 
K*12-l. D\-:(< Kx ): X lx.= x .x = b (4) 

h . (3) . (4) . 3 H :. yfr I (?a?)(0a?) . =* . -0- ! & : D . (ix)(<f>x) = b (5) 

h . (1) . (5) . D h . Prop 

It should be observed that we do not have 

(ix) (0a?) = 6 . = : -0 ! (ix) (<£#), D+.yjrlb 
for, if ~ E ! (ix) (<f>x), -0 ! (ix) (<f>x) is always false, and therefore 

-0 ! (ix) (<f>x) . D^ . 0- ! b 
holds for all values of b. But we do have 

*14171. I- :. (ix) (<f>x) = 6 . = : 0> ! 6 . D* . -0 ! (?a?) (<f>x) 

Bern. 
K*14\L7. 3 I- :. (ix) (0a?) = 6. D :-0!& . D^,. -0 ! (?a?)(0a?) (1) 

I- . #101 . *121 . D h :. f ! b . D,,, . 0< ! (?a?) (0a?) : D : 6 = 6 . D . (m?) (0a?) = 6 : 
[*1315] D:(?a?)(0a?) = & (2) 

h . (1) . (2) . D h . Prop 

*1418. H : . E ! (?a?) (0a?) . D : (a?) . -0a; . D . -0 (ix) (0a?) 

h . *101 . D H : (a?) . 0-a; . D . yfrb : 

[Fact] D I- :. 0a? . =3 . x = b : (x) . -0a? :D:<f>x.= x .x = b:yJrb: 

[*10-ll-28]D\-:.('3b):<j)X.= x .x = b: (a?).<0a?: D :(>&b) : <f>x .= x .x = b : yjrb:. 
[*10'35] DH ::(a&): <f)x.= x .x = b :.(x) . yfrxz.D : (g;6) : 0a? .=„ . x = b : yfrb :. 
[*14'1-11] D h :. E ! (?a?)(0a?) : (a?) . 0>a? : D : -0 (?a?) (0a?) :. D h . Prop 

The above proposition shows that, provided (ix) (0a?) exists, it has (speaking 
formally) all the logical properties of symbols which directly represent objects. 
Hence when (ix) (<f>x) exists, the fact that it is an incomplete symbol becomes 
irrelevant to the truth-values of logical propositions in which it occurs. 



SECTION B] 



DESCRIPTIONS 



181 



#14*2. H . (ix) (x-a) = a 
Dem. 
h . #14-101 . "5 h :. (ix) (x = a) = a. = : (g&) :x=*a.= x .x = b:b = a: 
[*13*195] =: x = a. = x .x = a (1) 

h . (1) . Id . Dh.Prop 

*14'201. H:E!(?a;)(^).D.(aar).^ 
Dem. 

h . #1411 . D h :. Hp . D : (g6) :<j>x.= x .x = b: 

[*10\L] D:(g&):#>. = .& = &: 

[*1315] D : (a&) . £6 : . D h . Prop 

#14 202. h :. <f>x . = x . x = 6 : = : (ix) (<f>x) = b : = : <f>x , =b x . b *= a? : = : b = (?a?) (<£#) 

h . #14"1 . D h :. (ix) (<j>x) = 6 . = : (gc) : (px . =« . ar = c : c = b : 
[*13195] = :<£#.=,,..#=&:. Dh.Prop 

[The second half is proved in the same way as the first half.] 

#14*203. h :. E ! (ix) (<f>x) . = : (ga?) . <f>x : <f>x . <f>y . D Xt y . x = y 

Dem. 
h . #14-12-201 . 3b:.El(ix)(<f>x).D:('zx).<f>x:<l>x.<f>y.3 Xty .x = y (1) 
h . #10'1 . D I- :. <f>b : <f>x . <f>y . x>y .x = y:D:<l>b:<f>x.<f>b.D x .x = b: 

[#533] 0:<j>b:<f>x.D x .x = b: 

[#13191] Di^&.D*.^: 

(f>% .D x .x = b: 
0:<f>x.= x .x = b (2) 



[#1022] 

I- . (2) . #10-1-28 . D h :. (g&):<£6 :<j>x.<f>y. D x>y .x = y:0 
[#10-35] Dh:.( a &).<£&:<^.<^. D„.* = y:D 

[#14-11] 3 

K(l).(3). Dh.Prop 

#14-204. h :. E ! (ix) (<f>x) . = : (g&) . (w) (0a>) = 6 
Z)em. 

h. #14-202. #10-11 . D 
h :. (6) :. <]>x . = x . x = b : = : (ix) (<f>x) = b :. D 
[*10-281] h :. ( 3 6) : <f>x . = x . x = b : = : (g&) . (?#) (^) = b 
h.(l).*14\Ll. Dh.Prop 
#14-205. h : yjr (ix) (<f>x) . = . (36) . b = (ix) ($x) . ^b [*14-202'1] 
#14-21. b:ylr(ix)(<f>x).O.El(ix)(<f>x) 
Dem. 

h . #14-1 . D 

h :. yfr {(ix) (<f>x)} . D : (36) : <f>x .== x .x = b:yfrb: 
[#10-5] D : (gi) :<l>x.= x .x = b: 

[#14-11] D : E ! (?#) (<£#) :. D h . Prop 



(fib) : <f>x . = x . x = b :. 
(Kb):<l>x.= x .x = b: 
El(ix)(<f>x) (3) 



(1) 



182 MATHEMATICAL LOGIC [PART I 

This proposition shows that if any true statement can be made about 
(ix) (fa), then (ix) (<f>x) must exist. Its use throughout the remainder of the 
work will be very frequent. 

When (ix) (<f>x) does not exist, there are still true propositions in which 
"(ix)(fa)" occurs, but it has, in such propositions, a secondary occurrence, 
in the sense explained in Chapter III of the Introduction, i.e. the asserted 
proposition concerned is not of the form yjr (ix) (fa), but of the form 
f\-ty (ix) (fa)}, in other words, the proposition which is the scope of (ix) (fa) 
is only part of the whole asserted proposition. 
#14-22. I- : E ! (ix) (fa) . = .<(> (ix) (fa) 

Dem. 

K #14122. D\-:.fa.= x .x = b:D.cf>b (1) 

1- . (1) . #4-71 . D b :. fa . = x . x — b : = : <j>x . = x . x = b : <f>b :. 

[*1011-281] D I- :. (g&) : fa . = x . x = b : = : (g&) : <f>x .= x .x= b : (f>b :. 

[*1411\L01] D h : E l(ix) (<}>x) . = .</> (ix) (fa) OK Prop 

As an instance of the above proposition, we may take the following: "The 
proposition ' the author of Waverley existed' is equivalent to 'the man who 
wrote Waverley wrote Waverley.'" Thus such a proposition as "the man 
who wrote Waverley wrote Waverley " does not embody a logically necessary 
truth, since it would be false if Waverley had not been written, or had been 
written by two men in collaboration. For example, " the man who squared 
the circle squared the circle" is a false proposition. 

*1423. V : E ! (ix) (fa . tyx) . = . <£ {(ix) (fa . tyx)) 

Dem. 
J- . #14-22 . D h :. E ! (ix) (<f>x . ^rx) . 

= : [(ix) (<f)X . yjrx)] : (f>{(ix) (<j>x . yjrx)} . sjr {(ix) (fa . -^rx)) 
[*10'5.*3-26] 3:<f>{(ix)(<f>x.yjrx)} (1) 

h . #14-21 O K </> {(ix) (<f>x . yfrx)] O . E ! (ix) (fa . ^x) (2) 

K (1) . (2) O K Prop 

Note that in the second line of the above proof #10-5, not only #3*26, is 
required. For the scope of the descriptive symbol (ix) (fa . tyx) is the whole 
product <£ [(ix) (<f>x . -yjrx)} . i/r {(ix) (fa . -fx)}, so that, applying #14*1, the 
proposition on the right in the first line becomes 

(g&) : fa . tyx .= x .x = b:<f>b. >Jrb 
which, by #10-5 and #326, implies 

(g&) : fa . yjrx . ~ x . x=b : <f>b, 
i.e. <j>{(ix)(fa.yjrx)}. 

#14-24. K . E ! (ix) (fa). = : [(ix) (fa)] = <t>V ■ =y ■ V = O) (fa) 

Dem. 
h . #141 . D I- : . [(ix) (fa)] :fa.= y .y = (ix) (fa) : 

= : (a 6 ) '.$y.= y .y = bi$y.= y .y = b: 



SECTION b] descriptions 183 

[*4-24.*10'281] s : (36) :<f>y .= y .y = b: 

[*14'11] = ■ E ! (?a?) (<M :• 3 K Prop 

This proposition should be compared with #14-241, where, in virtue of the 
smaller scope of (ix) (<f>x), we get an implication instead of an equivalence. 

#14-241. I- : . E ! (ix) (<f>x) . 3 : <f>y ■ = y • y = O) (<M 
Dem. 



K #14-203. 3h::Hp.3 

[Exp] 3 

[#1011-21] 3H::Hp.3 

[#471] 3 

[#13191] 

[#10-22] 

[#14-202] 



.<f>y.<f>x.D.y=x:. 
. (j>y . 3 : <j>x . 3 . y = a; " 
. (j>y . D : <f>x . "D x . y = x :. 
. <jty . = : <f>y:<]>x.D x .y = x: 

= :y = x.D x .<}>x:<f>x.'2 x .y = x: 

= :<f>x.= x .y = x: 

= :y= (ix) (<f>x) :: 3 K Prop 



#14-242. H :.<£#.=*.# = &: 3 :i|r&. = .^^)(<M [*1 4202-1 5] 
#14-25. H : . E ! (?a?) (<£#) . 3 : <]>x 3* yjrx . = . ifr (?#) (</>#) 

Dem. 
h . #4-84 . *10-27'271 . 3 H :: <j>x . = x . x = b : 3 :. <f>x 3* tyx . = : a? = 6 ■ >* ■ ^ : 
[#13-191] =.:^r6: 

[#14-242] =. -^(?a?) (</>«) (1) 

H . (1) . #1011-23 . 3 h :. (36) : <Jxb . =* . a? = b : 

3 : <£# D x -fx . = . i|r (?#) (<£#) (2) 
H. (2). #14-11. 31-. Prop 

#14-26. H : . E ! (ix) (<f>x) . 3 : (ga?) . <£# . ^ . = . -f {(m?) (<£#)} .~.<f>xD x fx 
Dem. 

K #1411. 3 

H:.Hp.D:(a6):^B.= x .af = 6 CO 

I- . #10-311 . 3 I- :: <j>x . = x . x = b : 3 :. <f>x . yjrx . ~ x . x = b . tyx :. 

[#10-281] 3 :. (gar) . 4>x . yjr x . = . (ga;) . a? = 6 . fx . 

[#13-195] =.yjrb. 

[#14-242] = ■ ^ {O) (<HJ ( 2 > 

I- . (2) . #10-11-23 . D 

I- :. (36) : <£# . =» . x = 6 : 3 : (g#) . <£# . \}rx . = . yjr {(?#) (<£#)} (3) 

1- . (1) . (3) . #1425 . 3 h . Prop 
#14-27. h :. E ! (ix) (4>x) . 3 : <f>x = x ^x. = . (ix) (<j>x) = (ix) (fx) 

Dem. 
J- . #4-86-21 . 3 b ::<£#. = . x = b : 3 :. <f>x . = . yjrx : = : -f x . = . x = b (1) 

h . (1) . #1011-27 . 3 I- :: </>*; .= x . x = 6 : D :. (*) :. <px . = . -fa; : = : ^ . = . as= b :. 
[#10-271] D-..(f>x.= x .ylrx: = :yfrx.= x .x = b: 

[#14-202] = : b = (ia?) (f ar) : 

[#14-242] =:(?a;)(<^) = (?«)(^) (2) 

I- . (2) . #1011-23 . #1411 OK Prop 



184 MATHEMATICAL LOGIC [PART I 

#14-271. b :. fa . = x ,yfrx : D : E ! (ix) (fa) . = . E ! (ix) (fx) 

Dem. 
b. #4*86. D b :: fa~ ^a?. D :. fa. = ,x = b : = : ^a;. = .# = & :: 
[#10-11-27] D H :: Hp . D :.(«):.</>«.= . a; = & : = : ^ . = . a? = 6 :. 

[*10'271] D : . («) : fa . = . # = 6 : = : (x) : ^a? . = . x = b :: 

[*1011-21] D h :: Hp . D :.(b) :. fa . = x . x = b : = : fx .= x . x = b :. 

[*10-281] 3:.fab):fa.= x .x = b: = :fab):+x.= x .x=bz: 

Dh.Prop 
#14272. b -.. fa . = x . yjrx :D : x (w)(fa) . = . x O) (^) 

Dem. 



h.*4'86. D h ::<£# = yfrx.D 
[*10ir414]Dh::Hp. D 

[Fact] D 

[#1011-21] Dh::Hp. D 

[#10-281] D 



: . fa ,=.x=bi=: yjrx . = . x=b : . 
:. <£a;. =^.a;=6 : = : -^ra;. = x .x = b :. 
:.fa.= x .x=b:x^>- = ' ^ x . =* . a; = 6 : ^6 :. 
: . (b) : . 0a; . = x . a; = b : ^6 : = : yjrx .= x .x=b:xb:. 
:. fab) : fa . = x . x = b : xb : = 

: fab) :-<lrx.= x .x = b:xb:. 
[*14'101] D : . x (ix) (fa) . = . x (ix) (fx) ::Db. Prop « 

The above two propositions show that E ! (?#) (<£a;) and x( lx )(fa) are 
" extensional" properties of fa, i.e. their truth-value is unchanged by the 
substitution, for fa, of any formally equivalent function yfrft. 

*14'28. b:El(ix)(fa).~.(ix)(fa) = (ix)(fa) 

Dem. 
b . #1315 . #4-73 . D b :. fa . = x . x = b : = : fa . = x . x = 6 : b = 6 (1) 

K(i).*ioir28i.:> 

r-:.(g6):<^a;.= a ..a;=6: = :(a6):0a;.= a ;.a:=6:6=6 (2) 
l-.(2).*14-lll.Dh.Prop 

This proposition states that (ix) (fa) is identical with itself whenever it 
exists, but not otherwise. Thus for example the proposition "the present 
King of France is the present King of France " is false. ' 

The purpose of the following propositions is to show that, when El(ix)(fa), 
the scope of (ix) (fa) does not matter to the truth-value of any proposition 
in which (ix) (fa) occurs. This proposition cannot be proved generally, but 
it can be proved in each particular case. The following propositions show 
the method, which proceeds always by means of #14-242, #10-23 and #14*11. 
The proposition can be proved generally when (ix)(fa) occurs in the form 
X ( lx ) (fa)* an d x ( qx ) ( fa) occurs in what we may call a " truth-function," i.e. 
a function whose truth or falsehood depends only upon the truth or falsehood 
of its argument or arguments. This covers all the cases with which we are 
ever concerned. That is to say, if x (ix) (fa) occurs in any of the ways which 
can be generated by the processes of #1 — #11, then, provided E ! (ix) (fa), 
the truth-value of/ {[(ix) (fa)] >x(ix)(fa)} is the same as that of 

[(ix)(fa)].f{ x (ix)(fa)}. 



SECTION b] descriptions 185 

This is proved in the following proposition. In this proposition, however, the 
use of propositions as apparent variables involves an apparatus not required 
elsewhere, and we have therefore not used this proposition in subsequent 
proofs. 

*W3. \-z.p = q . D p , q .f(p) =f(q) : E ! (70) (00) : 3 : 

f{\(ix) (00)] . x (70) (00)} . = . [(ix) (*»)] .f{ x (ix) (00)} 
Dem. 

h . *14'242 . D 

r- : . <f>x . = x . x = b : : [(70) (00)] . % (70) (00) . = . X 6 (1) 

h.(l).DH:.p S 5.D M ./(p)=/(j):^c.5..«6:D:. 

/{[(w) (Ml -xO*)(M • = •/(%*) (2) 
h . *14-242 . 3 

h :. 00 . = x . x = b : D : [(10) (0*)] ./{% (w) <**)} - s ./(*&) (3) 

r.(2).(3).Z> 

I- :.p = ? . D P)9 ./(^)=/(^) : 00 . =0, . = 6 O : 

/{[(**) (00)] . x (m) (**)} ■ s . [(10) (00)] .f\x (w) (**)} (4) 

I- . (4) . *10'23 . *14 11 . D h . Prop 

The following propositions are immediate applications of the above. They 
are, however, independently proved, because *14 - 3 introduces propositions 
(p, q namely) as apparent variables, which we have not done elsewhere, and 
cannot do legitimately without the explicit introduction of the hierarchy of 
propositions with a reducibility-axiom such as *121. 

*14'31. H : : E ! (70) (00) . D : . [(?0) (00)] . p v x (*#) (<H • 

= : p . v . [(?0) (00)] . x (™) (<H 
Dem. 
h .*14-242 . D h i.<f>x ._= x .x = b: D :[(70)(00)]./>v%(70)(00). = .pvx& (1) 
I- . *14'242 . D H :. 00 . = x . x = b .O : [{ix) (00)] . x i™) (<K> ■ = ■ X b '• 
[*4'37] D : p v [(70) (00)] X («0 (<H • s . ;> v x h (2) 

h . (1) . (2) . D I- :. 00. = x .x = b : Z> : [(70) (00)] . p v * (70) (00) . 

= . p v [(70) (00)] X (70) (00) (3) 
h . (3) . *10-23 . *1411 .Dh. Prop 

The following propositions are proved in precisely the same way as *14'31 ; 
hence we shall merely give references to the propositions used in the proofs. 

*14-32. h : . E ! (70) (00) . = : [(70) (00)] . ~ X ( lx ) (<K> ■ 

= . ~ {[(70) (00)] . X (M?) (<f>®)} 

[*1 4242 . *411 . *1023 . *14'11] 

The equivalence asserted here fails when ~ E ! (70) (00). Thus, for example, 
let <f>y be " y is King of France." Then (70) (00) = the King of France. Let 
XV be "y is bald." Then [(70) (00)] .~%(70)(00) . = . the King of France 
exists and is not bald ; but ~ {[(70) (0#)] • X ( lx ) (4*°°)} ■ = ■ it is false that the 
King of France exists and is bald. Of these the first is false, the second true. 



186 MATHEMATICAL LOGIC [PART I 

Either might be meant by " the King of France is not bald," which is am- 
biguous ; but it would be more natural to take the first (false) interpretation 
as the meaning of the words. If the King of France existed, the two would be 
equivalent ; thus as applied to the King of England, both are true or both false. 

*14-33. h : : E ! (ix) (fa) . D : . [(?*) (fa)] .p0% ( lcc ) (<H - 

=:p.D. [(ix) (fa)] ■ X ( lx ) (4> x ) 
[*14242 . *4-85 . *1023 . *14-11] 

*14-331. V :: E ! (ix) (fa) . D :. [(ix) (fa)] . % (ix) (fa) Op . 

= : [(ix) ((fix)] . x X ) (<\> x ) -3-P 
[*4-84 . *14'242 . *1023 . *1411] 

*14 332. h :: E ! (ix) (fa) . D :. [(ix) (fa)] .p = % (™) (4*) ■ = 

: p . = . [(ix) (fa)] . x (ix) (fa) 
[*4-86 . *14-242 . *1023 . *1411] 

*14'34, t- : . p : [(ix) (fa)] . X (™) (fa) = = ■ [(**) (fa)] -P-X (»*) (4> x ) 

This proposition does not require the hypothesis E ! (ix) (fa). 

Devi. 
V . *14 1 . D 

I- :.p : [(ix)(fa)] .x(w)(fa) : = :p: (a&) : fa.= x .x = b : %& : 
[*10'35] = : (g&) : p : fa .= x .x = b : %6 : 

[*14*1] = : [(7«) (fa)] :p.x (i x ) (<t> x ) '-Oh . Prop 

Propositions of the above type might be continued indefinitely, but as they 
are proved on a uniform plan, it is unnecessary to go beyond the fundamental 
cases of p v q, ~p, p D q and p . q. 

It should be observed that the proposition in which (ix) (fa) has the 
larger scope always implies the corresponding one in which it has the smaller 
scope, but the converse implication only holds if either (a) we have E ! (ix) (fa) 
or (b) the proposition in which (ix) (fa) has the smaller scope implies 
E ! (ix) (fa). The second case occurs in #14'34, and is the reason why we 
get an equivalence without the hypothesis E I (ix) (fa)-. The proposition in 
which (ix) (fa) has the larger scope always implies E ! (ix) (fa), in virtue of 
♦14-21. 



SECTION 

CLASSES AND RELATIONS 
*20. GENERAL THEORY OF CLASSES 

Summary o/#20. 

The following theory of classes, although it provides a notation to represent 
them, avoids the assumption that there are such things as classes. This it does 
by merely defining propositions in whose expression the symbols representing 
classes occur, just as, in #14, we defined propositions containing descriptions. 

The characteristics of a class are that it consists of all the terms satisfying 
some propositional function, so that every propositional function determines a 
class, and two functions which are formally equivalent {i.e. such that whenever 
either is true, the other is true also) determine the same class, while conversely 
two functions which determine the same class are formally equivalent. When 
two functions are formally equivalent, we shall say that they have the same 
extension. The incomplete symbols which take the place of classes serve the 
purpose of technically providing something identical in the case of two functions 
having the same extension ; without something to represent classes, we cannot, 
for example, count the combinations that can be formed out of a given set of 
objects. 

Propositions in which a function <£ occurs may depend, for their truth- 
value, upon the particular function ^, or they may depend only upon the 
extension of <f>. In the former case, we will call the proposition concerned an 
intensional function of </>; in the latter case, an extensional function of <£. 
Thus, for example, (x) . <f>x or (g<c) . <f>x is an extensional function of <fy, 
because, if <f> is formally equivalent to yjr, i.e. if (fix . = x . yfrx, we have 
(x) . (fix . = . (x) . tyx and (g#) . <f>x . = . (go?) . tyx. But on the other hand 
" I believe (x) . <f>x " is an intensional function, because, even if <f>x.= x . yfrx, 
it by no means follows that I believe (#) . ^rx provided I believe (x) . <f*x: The 
mark of an extensional function/ of a function <j> ! 2 is 

<f> \x .= x .^lx: D M :/(<£ ! 3) . = ./(^ ! S). 
(We write "<f> ! £" when we wish to speak of the function itself as opposed to 
its argument.) The functions of functions with which mathematics is specially 
concerned are all extensional. 

When a function of <$> ! z is extensional, it may be regarded as being 
about the class determined by <f> 1% since its truth- value remains unchanged 
so long as the class is unchanged. Hence we require, for the theory of classes, 
a method of obtaining an extensional function from any given function of a 
function. This is effected by the following definition: 



188 MATHEMATICAL LOGIC [PART I 

#20-01. f fi^z)}. = :(<&<!>): <f>lx.= x .fx:f{<f>l$} Df 
Here / {£ ($z)) is in reality a function of yfrz, which is defined whenever 
/ {<£ ! z) is significant for predicative functions <f> ! t. But it is convenient to 
regard / {£ (yfrz)} as though it had an argument £ (yfrz), which we will call 
"the class determined by the function yfrz." It will be proved shortly that 
/ {z (yfrz)} is always an extensional function of yfr%, and that, applying the 
definition of identity (*13-01) to the fictitious objects z (<f>z) and z (yfrz), we 
have 

z ((f>z) = z (yfrz) . = : (x) : <$>x . = . yfrx. 
This last is the distinguishing characteristic of classes, and justifies us in 
treating z (yfrz) as the class determined by yfr%. 

With regard to the scope of 2 (yfrz), and to the order of elimination of two 
such expressions, we shall adopt the same conventions as were explained in 
#14 for (ix) (<f>x). The condition corresponding to 

E ! (ix) (yfrx) is (a<£) : <f> I x . = x . yfrx, 
which is always satisfied because of #12*1. 
Following Peano, we shall use the notation 

x e z (^frz) 
to express "x is a member of the class determined by yfrz." We therefore 
introduce the following definition: 
#2002. x e (0 ! 2) . = . <f> ! x Df 

In this form, the definition is never used ; it is introduced for the sake of the 
proposition 

H :. x e 2 (yfrz) . = : (g<£) : ^y . = y . <f> ! y : <\> ! x 

which results from #20'02 and #20-01, and leads to 

h : x e z (yfrz) . = . yfrx 
by the help of #1 2*1. 

We shall use small Greek letters (other than e, i, it, (f>, yjr, %, 6) to represent 
classes, i.e. to stand for symbols of the form 2 (<f>z) or z (<f> ! z). When a small 
Greek letter occurs as apparent variable, it is to be understood to stand for a 
symbol of the form £ (<f> ! z), where <j> is properly the apparent variable con- 
cerned. The use of single letters in place of such symbols as z (<f>z) or z (<f> ! z) 
is practically almost indispensable, since otherwise the notation rapidly becomes 
intolerably cumbrous. Thus "x e a" will mean "x is a member of the class a," 
and may be used wherever no special defining function of the class a is in 
question. 

The following definition defines what is meant by a class. 
#20-03. Cls = o {( a <£). a = z(<f> ! z)} Df 

Note that the expression "a {(g<£). a = 2(0 ! z)}" has no meaning in 
isolation: we have merely defined (in #20*01) certain uses of such expressions. 
What the above definition decides is that the symbol "Cls" may replace the 
symbol "a {(g<£) . a = z (<f> ! *)}," wherever the latter occurs, and that the 



SECTION C] GENERAL THEORY OE CLASSES 189 

meaning of the combination of symbols concerned is to be unchanged thereby. 
Thus "Cls," also, has no meaning in isolation, but merely in certain uses. 

The above definition, like many future definitions, is ambiguous as to 
type. The Latin letter z, according to our conventions, is to represent the 
lowest type concerned ; thus <f> is of the type next above this. It is convenient 
to speak of a class as being of the same type as its defining function ; thus a 
is of the type next above that of z, and "Cls " is of the type next above that 
of a. Thus the type of " Cls " is fixed relatively to the lowest type concerned ; 
but if, in two different contexts, different types are the lowest concerned, the 
meaning of "Cls" will be different in these two contexts. The meaning of " Cls" 
only becomes definite when the lowest type concerned is specified. 

Equality between classes is defined by applying #13*01, symbolically un- 
changed, to their defining functions, and then using #20*01. 

The propositions of the present number may be divided into three sets. 
First, we have those that deal with the fundamental properties of classes ; 
these end with #20-43. Then we have a set of propositions dealing with both 
classes and descriptions; these extend from #20*5 to #20*59 (with the ex- 
ception of #20-53 54). Lastly, we have a set of propositions designed to prove 
that classes of classes have all the same formal properties as classes of in- 
dividuals. 

In the first set, the principal propositions are the following. 
#2015. h :. yjrx . = x . %? : = . z ($>z) = z (x z ) 

I.e. two classes are identical when, and only when, their defining functions 
are formally equivalent. This is the principal property of classes. 

#20-31. H : . z (^z) = z (xz) .=:xez (yjrz) .= x .xez ( X z) 

1.0. two classes are identical when, and only when, they have the same 
members. 

#20-43. \-:.a = p. = :x6a.= x .x€@ 

This is the same proposition as #20*31, merely employing Greek letters 
in place of z (\Jrz) and z (xz). 

#20-18. h :. 2 (<f>z) = 2 (fz) .D:f{z (<j>z)} . = ./{z (yjrz)} 

I.e. if two classes are identical, any property of either belongs also to the 
other. This is the analogue of #13 - 12. 

#202 2122, which prove that identity between classes is reflexive, symmetrical 
and transitive. 

#20-3. h:xez (■fz) . = .^x 

I.e. a term belongs to a class when, and only when, it satisfies the defining 
function of the class. 

In the second set of propositions (#20-5 — '59), we show that, under suitable 
circumstances, expressions such as (ix) (<j>x) may be substituted for x in #20 # 3 . 



190 MATHEMATICAL LOGIC [PART 1 

and various other propositions of the first set, and we prove a few properties 
of such expressions as "(?a) (fa)," i.e. " the class which satisfies the function/." 
Here it is to be remembered that "a" stands for "2(<f>2)" and that "fa" 
therefore stands for " f {2 (<f>e)\." This is, in reality, a function of <pz, namely 
the extensiOnal function associated with f(yjrlz) by means of #2001. Thus 
an expression containing a variable class is always an abbreviation for an 
expression containing a variable function. 

In the third set of propositions, we prove that variable classes satisfy all 
the primitive propositions assumed for variable individuals or functions, whence 
it follows, by merely repeating the proofs of the first set of propositions (#20'1 
-43), that classes of classes have all the formal properties of classes of in- 
dividuals or functions. We shall never have occasion explicitly to consider 
classes of functions, but classes of classes will occur constantly — for example, 
every cardinal number will be defined as a class of classes. Classes of relations, 
which will also -frequently occur, will be considered in #21. 

#2001. f{z(^z)}. = :(^)'.^\x.= x .^rx'.f{j>\z} Df 
#20*02. x e (<f> I z) . = . ! x Df 

#2003. Ob = 3. {<a0). a •=*($!*)} Df 

The three following definitions serve merely for purposes of abbreviation. 
*20'04. x,yea. = .x€a.yea Df 

#2005. x, y,zea. = .x, yea.zea Df 
#2006. x~ea . = . ~(xea) Df 

The following definitions merely extend to symbols representing classes 
the definitions which have already been given for other symbols, with the 
smallest possible modifications. 

#20-07. (a).fa. = .(<j>).f{z(<t>lz)} Df 

#20-071. (go) ./«. = . (30) •/(* (0 ! *)} Df 

#20-072. [(?a) (<£«)] ■/<» <0«) ■ = : (a*/) : 0« ■ =• • « = 7 : fj Df 
#2008. f{ot (<f a)l . = : (30) : fa . =. . <f> ! a :/(0 I &) Df 

#20081. ae i|r ! & . = . -«|r ! a Df 

The propositions which follow give the most general properties of classes. 
#20-1. r :./{$(**)} • s : <30) :<\>\x .= x .tyx:f{4>\$] [*4-2 . (#20-01)] 

#20-11. I" : . fx . = x . x x '■ D : / (2 ( Wl ■ = • / I* (X*)) 

Dem. 
\- . #4-86 . h :: Hp . D :. I as . = x . fx : =+ : <f> ! x . = x . x ® '- 
[#4-36] D:.<f>lx.= x .fx :/{0 ! z] : =* : <f> ! as . = x . X x = /{0 } - *} '•• 

[#10-281] 3 :. (g0) : lx.= x ,. -fx :/{<f>.l z) : 

= : (g0) :</>! x. =.. X x :/[0 lz}:. 
[#20-11 O :.f{z(irz)\ . ee .f{z( X z)} -=>■-■ ^op 



SECTION C] GENERAL THEORY OF CLASSES 191 

This proves that every proposition about a class expresses an extensional 
property of the determining function of the class, and therefore does not 
depend for its truth or falsehood upon the particular function selected for 
determining the class, but only upon the extension of the determining function. 
*20-lll.hi.f($l2).=t.g(<f>l2):3:f{2(<j>lz)}.=t.g{2($lz)} 

Dem. 
KFact. Dbi:np.D:.<f>lx.= x .^lx:f(^lz): = :<f>lx.= x .ylrlx:g(yfrlz):i 

[*10281] D:.(a^)r^!«.= jB ^!*:/(^!t): = :(a^r):^t*.s«.^!a?:flr(^!S):, 

[#201] D :./{*(* !*)} - = -9 {z(<f> I *)} (1) 

r- . (1) . *1011-21 . D h . Prop 

♦20112. r :. to) v.f{2 (<£ ! *)} • ^ - g I {z (<f> I *)} 

Dem. 

h . #121 . 3 h :. ( a5 r) :/(<£ ! 2) . =* . £ ! (<f> I *) (1) 

K (1) . #20111 . 3 1- . Prop 

Thus the axiom of reducibility still holds for classes as arguments. 
#2012. I- : (g<£) : <£ ! x . = x . ^x :f{z (-fz)} . = ./{z (<f>l z)} [#2011 . #121] 
#2013. f- : . -fyx . =a, . %ar : 3 . 2 (^) = 2 fo*) 

The meaning of " z (^jrz) — z (xz) " is obtained by a double application of 
#2001 to #1301, remembering the convention that 2(tyz) is to have a, larger 
scope than 2 (%z) because it occurs first. 

Dem. 
H . #20 1 . 3 h ; : 2 (yjrz) = 2 ( X z) . s : . (g<£) : ^a; . ■=« . £ ! a; : <£ ! 2 = 5 (^^) : . 
[#201] = :.(^4>,0);^x.= x .<f>\x:xx^ x .nx:<f>l2 = 0l2 (1) 

I- . #121 . #10-321 . 3 

h :: Hp . 3 :. (g$>) : -^a- . = x . <f> l& : yx .= x .$>\x\. 

[#13195] 3 :. (a0, 0) :. tyx. = x .<j>lx: x x .= x .0lx:<j>l2 = 0l2 (2) 

f- . (1) . (2) . 3 h . Prop 
#2014. h : . 2 (ifrs) = £ (^) . 3 : tyx . = x . x x 

Dem. 
h . #201 . 3 h :: 2(^s)«2(#er) . = :. (g£) : ^a?. =,..^ ! a?: <f>l2 = 2( X z) :. 
[*20l] = :. (a<£, 0):.^x.= x .<f>lxzxx.= x . 01x: $12=0 ! 5 :. 
[#13195] = :. (g</>) :, yjrx . = x . <£ ! x : x x • ■=* • <£ ! * : - 
[#10322] 3 .:. i^a; . = x . ^ " ">*" • Prop 

This proposition is the eou verse of #2013. 
#2015. \-:.fx.= x .xxz = .z(^z) = z(xz) [#20-1314] 

This proposition states that two functions determine the same class when, 
and only when, they are formally equivalent, i.e. are satisfied by the same set 
of values. This is the essential property of classes, and gives the justification 
of the definition #2001. 



192 



MATHEMATICAL LOGIC 



[PART I 



#20151. h . (g<£) . 2 (^z) = £ (0 ! z) 

Bern. 
>.*2015. 3h:.ylrx.= x .<f>lix::D.'z*(ylrz) = 2(<f>lz):. 
[*10-ll-28] Db :.(%<!>): fx.= x .<f>lx:D.(K<f>).2(yfrz) = 'z(<f>lz) (1) 
J-. (1).*1 2-1. Dh. Prop 

In virtue of this proposition, all classes can be obtained from predicative 
functions. This fact is especially important when classes are used as apparent 
variables. For in that case, according to the definitions #2007071, the ap- 
parent variable really involved is a predicative function. In virtue of #20 - 151, 
this places no limitation upon the classes concerned, except the limitation 
which inevitably results from the nature of their membership. A class, there- 
fore, unlike a function, has its order completely determined by the order of 
its possible members, i.e. of the arguments which render its defining function 
significant. 

*2016. h : (a</>) iff (irz)} . = ./{2 (<f> ! z)) [*20'12] 

*2017. h : (<£) .f{H<f> ! *)} ■ 3 ./{2 (yfrz)} [*20'16 . #101] 

#2018. h:.2(<K> = *0K>-3:/ WM "• s ./{*(*«)} [*201115] 
*2019. h :. $ {&£) = z { X z) . = :(/) if I 2 (**) . D ./! S (%*) 

Dem. 
\- .#2018 .*1011'21 . D h :.$We) = 2(x*) • ? : 

(/):/!*(**). D./!*(x*) (1) 
h .*2018\L5 .Db::<j>lw.= x .yfra;:0la;.= x .x^ :f\%(y\rz).D .f\z\x z ) • D : 

/!*(*!*). D./!*<0!*) (2) 
I- . (2) . *10-ll-27-33 . D 

h::^!a;.s..^:^!*.s,. X *-:.(/):/Ii(^).D./!^(x*):-3» 

[*20-112.*10-l] D :.<£!#.=*.<£! a : D :<£!#.=*. !« :• 
D :.<£!#. =*.#!#:. 

D :. 2 (>K> = *(%*) (3) 

l-.(3).*10-ll-23-35.D 

h::(a^,^:^!*.s..^:^!*.s..X*:.(/):/^(^)-3-/^(X*)» 

D.*'W*)-S( X *) (4) 

h.(4).*12-l.Dh:.(/):/!^(^).D./!t(^):3.t(^) = a(^) (5) 

h . (1) . (5) . DH.Prop 

#20 191. h :. 2(^) = * O) • = :(/) :/! £ ty*) . s ./! 2( X *) 

[#20-18-19 . #10-22] 
#20-2. h . 5 (<f>z) = 2 (<£*) 

h . #20-15 . D H :. 2 (<f>z) = z((f>z) . = z <f>x . =„ . <f>x (1) 

K (1) . #42 . #10-11 . D r- . Prop 



[#4-2] 

[#10301-32.Hp] 

[#2015] 



SECTION C] 



GENERAL THEORY OP CLASSES 



193 



#2021. h :2(^)-=2(^s). = . 2 (^*) = $(£*) [#2015 . #1032] 

#2022. h : z (<f>z) - 2 (^) . 2 (<fz) = 2 ( % *) . D . 2 (£*) = a ( % *) 

[*2015 . #10-301] 
The above propositions are not immediate consequences of *13-151617, 
for a reason analogous to that explained in the note to #14*13, namely because 
f{z (<f>z)} is not a value of fx, and therefore in particular " z {<\>z) - z (^rz) " is 
not a value of " x = y" 

#20-23. \-:z(<f>z)^^(^z).^(<f>z) = ^( X z).D.z(yfrz) = ^( X z) [#20-21-22] 
#20-24. b:2(ylrz) = z-(<j>z).z'( X z) = %(<l>z).D. / z(^z) = 2( X z) [*20'21-22] 
#20-25. h :.a = z((j)z).= a . ct = 2tyg) : = .z(cf>z) = z(fz) 

Dem. 
h . #10-1 . D h :. a = z(<f>z) .= a .a = 2(yjrz) : D : 

z(<j>z) = 'z((f>z). = .^(<j>z) = ^(fz): 
[*20'2] D:^(<f>z) = ^(yjrz) (1) 

b . #20-22 . D \- : a = £ (<£*) . £-(<^) = £ (^) . D . a = £ (yfrz) : 
[Exp.Comm]Dh:.a(^) = a(^).D:a = ^(^).D.a = t(^). (2) 

h. #20-24. 3 h :.$ (<}>z) = z (yfrz) . a = z (yjrz) .D . a = 2 (cf>z) :. 
[Exp] D h :. %{<j>z) = 2{fz) . D : a = £(^) . D . a = £ (£?) ( 3 ) 

h.(2).(3). Dh:J(^) = ^(^).D:a = t(^). = .a = J(^) : . 
[*10-11-21] Dl-:.f(^) = t(^).D:a=f(^).=„.a = t(^) (4) 

l-.(l).(4). Dh.Prop 
#20-3. \-;xe%{tyz). = .tyx 
Dem. 

h. #201.3 



r- : : x e z {*\rz) . = 
[(#20-02)] = 
[#10-43] = 

[#10-35] .= 

[#121] = 



• (a</>) -iry .= y .<\>\y.^x -.. 

.(a<^>): -sjry.^y.^ly^yjrx:. 
:. yjrx:: D b . Prop 

This proposition shows that x is a member of the class determined by yjr 
when, and only when, x satisfies yjr. 

#20-31. h :. z (yjrz) = 2 ( X z) .= :xez (fz) .= x .xez ( X z) [#20-15-3] 
#20-32. \-.x~{x6z-(<!>z)}=$(<l>z) [#20-3-15] 

#20-33. b:.a = 2(<f>z). = :x€a.= x .(l>x 
Dem. 

h. #20-31. ^\-:.a = ^(<f)z). = :x6a.= x .X€^(cf>z) (1) 

h. (1). #20-3. DK Prop 
Here a is written in place of some expression of the form z tyz). The 
use of the single Greek letter is more convenient whenever the determining 
function is irrelevant. 

R&W I jo 



194 MATHEMATICAL LOGIC [PART I 

#20*34. [~:.cc — y.='.x€a.Da-yea 
Dem. 

h . #4*2 . (#20-07) . D I- : . x e a . D a . y e a : = : x e Z (<f> ! z) . D* . y e 2 (<f> I z) : 
[#20*3] =:</>!#. D^.^ly: 

[#13*1] =:x = y:.Db.Frop 

The above proposition and #20*25 illustrate the use of Greek letters as 

apparent variables. 

#2035. \-:.x = y. = :xea.= a .yea [#20*3 .#13-11] 

#20 4. h : a e Cls . = . ( 3 0) . a = %(<]> 1 z) [*20"3 . (#20-03)] 

*2041. h.f(^)eCls [*20'4-151] 

#2042. H . z (z e a) = a 

A Greek letter, such as a, is merely an abbreviation for an expression of 

the form 2 (<f>z) ; thus this proposition is #20*32 repeated. 

Dem. 

h . #20-3 . #10-11 . D I- :x eztyz) . ~ x . fx : 

[#20-1 5] D h . £ [x e z tyz)} = £ (^#) . D h . Prop 

#20-43. h :. a = /3. = :#€«.=*. #e/3 [#2031] 

The following propositions deal with cases in which both classes and 
descriptions occur. In such cases, we shall, in the absence of any indication 
to the contrary, adopt the convention that the descriptions are to have a 
larger scope than the classes, in applying the definitions #14-01 and #2001. 

#20-5. h : (ix) (<f>x) e z (^z) . = .^{(ix) (<■*«)} 
Dem. 



■ (a c ) '• 4>x .= x .x = c:cez (yjrz) :. 
. (go) : <f>x . = x . x = c : tyc :. 
.yfr{(ix)(<f)x)} ::DK Prop 



h . #14-1 . D h :: (iae) (<f>x) e z (tyz) . = 

[#203] = 

[#14-1] = 

#20-51. r- :. (7a?) (<jix) = b. = : (ix) (<f>x) ea.=..6ea 
Dem. 
h . #20-5-3 . D 

h :. (ix)(<j>x) ez(yfr I z) . = .b ez(yfrl z) : = : ^jrl (ix) (<j>x) . = . <f ! b :. D 
[#10-11] h :. (las) (<j>x) e a . =« . & e a : = : f ! (far) (<K> . = + . -f ! 6 : 
[#14-17] = : ('«) (£*) = & :■ 3 r- . Prop 

*2052. h : . E ! (i«) (</>#) . = : (a&) : O) (<H e a . =« . 6 e a 

K #20 51. #1011-281.3 

h :. (g&) . (ix) (<\>x) = b . = : (a&) : (ix) ($x) e a . = a . & e a (1) 

h.(l). #14-204. Dh. Prop 

#20-53. l-:./3 = a.D0.</>/3: = .</>a 
This is the analogue of #13191. 



SECTION C] 



GENERAL THEORY OF CLASSES 



195 



Dem. 

K*101. Dh:./3 = a.D0.0/3:D:a = a.O.0a: 
[*20-2] D:0a (1) 

h . *2()-18-21 . D h :. /3 = a . D : 0a . D . 0/3 :. 
[Comm] DH:.0a.:>:/3 = a.D.0/3:. 

[*1011-21] D\-:.(f>a.D:/3^a.D fi .(f>^ (2) 

h . (1) . (2) . DKProp 
#2054. h : {ft/3) . /3 = a. . 0/3 . = . 0a 

This proposition is the analogue of #13195. 
Dem. 

h . #2018 . #1011 . D I- : /3 = a . 0/3 . Dp. 0a : 
[#10-23] Dh:(g/3)./3 = a.0/3.D.0a (1) 

K #20'2 . #32 . D I- : 0a . D , a = a . 0a . 
[*10-24] D.( a /3)./3 = a.0/3 (2) 

K(l).(2). Dh.Prop 

#20 55. I- . z (0*) = (?a) (xea.= x . <f>x) 
Dem. 

h . #2033 . D h :: x e a . =* . <f>x : = a . a = z (j>z) :. 
[*20-54] Dh:.(Rl3)i.X6a.= x .cj>x:= a .a = l3:.z(<l>z) = {3:. 
[*14-1] D h . £ (00) = (7«) (x e a . = x . <f>x) . D h . Prop 
*20'56. h . E ! (?a) (^o.e,. 0a;) [#2055 . *14*21] 
*2057. h :. 2 (00) = ( 7 a) (fa) . D : g {z (00)} . = .g {(»«) (/a)} 

.Dem. 
h . #141 . D h :: Hp . = :. ( a /3) : /a . = a . a = /3 : (00) = £ :. 
[*20-54] = :./a.= a .a = £(00) (1) 

K*141. 0\-:.g{(ia)(fa)\.^:(^):fa.= a .a = /3:g^ (2) 

h.(l).(2).3H::Hp.D:. fi r{(ia)C/a)}.= 
[*13183] = 

[#2054] = 

#2058. K 2(00) = (7a) [a = £(00)} 
Dem. 

h . *4-2 . #10-11 . D h : a = f(00) . = a .a = z(<j,z) : 
[#20-54] D I- :. ( a /3) : . a = £ (0*) . = a . a = £ : 2 (0s) = /3 :. 

[#141] D h . $ (0^) = ( ?a ) {« = 2 (0*)} . D h . Prop 

#20-59. h : 2 (0^) = (id) (fa) . = . (la) (fa) = z(cf>z) 
Dem. 



(>&&):a = z( ( j>z).= ll .a = {3:g/3: 
( 3 /3).£(00) = /3.#/3: 
g{z(<f>z)} ::Dh. Prop 



f- . #20-1 . D h :. z (00) = (?a) (/«) . = 
[#1413] = 

[#201] = 



(ftyjr) :^>x.= x ^lx:yjrlz=: (7a) (fa) : 
(g^r) : 0a; . =, . i/r ! # : (7a) (fa) = i/r ! 2 : 
(7a)(/a) = 2(00):.Dh.Prop 

13—2 



196 MATHEMATICAL LOGIC [PART I 

In the following propositions, we shall prove that classes have all the 
formal properties of individuals, and have the same relations to classes of 
classes as individuals have to classes of individuals. It is only necessary to 
prove the analogues of our primitive propositions, and of our definitions in 
cases where their analogues are not themselves definitions. We shall take 
the propositions #10 Iiri212ri22, rather than those of #9, and we shall 
prove the analogue of #10-01. As was pointed out in #10, we shall thus have 
proved everything upon which subsequent proofs depend. The analogues of 
#200102 and of #14-01 remain definitions, but those of #1001 and #13-01 
become propositions to be proved. #9'131 must be extended by the definition: 
Two classes are "of the same type" when they have predicative defining 
functions of the same type. In addition to these, we have to prove the 
analogues of #10 riri2121122, #1107 and #12-1-11. When these have been 
proved, the analogues of other propositions follow by merely repeating previous 
proofs. These analogues will, therefore, be quoted by the numbers of the 
original propositions whose analogues they are. 

#20-6. h:(aa)./a. = .~{(a).~/a} 

Dem. 

h . #4-2 . (#20-071) . 3 

h:(aa)./a. = .(a0)./{3(0!*)}. 
[(#10-01)] = -~[(*).~/{*(* **)}]. 
[(#20-07 )] = . ~ {(a) . <-/«} OK Prop 

This is the analogue of #10-01. 
#20-61. h :(«)./«. D.//3 

Dem. 

h . #10-1 . (#20-07) O I- : (a) ./a . D ./{z (<f> I z)) : D h . Prop 

This is the analogue of #10*1. 

In practice we also need 

r:(«)./«.D./{£0K>}. 
This is #20-17. 

We need further h . (got) . z (yfrz) = a. 

This is #2041. 
#2062. When //3 is true, whatever possible argument of the form z(<j>lz) 
/3 may be, then (a) ./a is true. 

This is the analogue of #1011. 
Dem. 

h . #10-11 . D . when f[z (<j> ! z)) is true, whatever possible argument <j> may 
be, then (cf>) ./{z((f> ! z)} is true, i.e. (by *20'07), (a) ./a is true. 

#20-63. \-:.(a).pvfa.D:p.v .(a).fa 
This is the analogue of #10-12. 



SECTION C] GENERAL THEORY OF CLASSES 197 

Dem. 

h . #4*2 . (#20*07) . D 

h:.(a).j>v/«. = :($).pv/{*(*!*)}: 

[#1012] = : p. v. (£)./{$ (*!*)}: 

[(#20*07)] = : p . v . (a) ./a : . D H . Prop 

#20*631. If "fa." is significant, then if /3 is of the same type as a, " fft" is 
significant, and vice versa. 

This is the analogue of #10121. 

Dem. 

By #20151, a is of the form z {$\z), and therefore, by #20*01, fa is a 
function of $ ! 2. Similarly /3 is of the form z (yjr ! z), and ffi is a function of 
■«/r ! z. Hence by applying #10121 to </> ! z and t/r ! 2 the result follows. 

#20*632. If, for some a, there is a proposition fa, then there is a function fa, 
and vice versa. 

Dern. 

By the definition in #20*01, f{z (yjr ! ^)] is a function of i/r ! z . Hence the 
proposition follows from #10*122. 

#20*633. "Whatever possible class a may be, /(a, /3) is true whatever possible 
class /3 may be" implies the corresponding statement with a and /3 inter- 
changed except in "/(a, fi)." 

This is the analogue of #11*07, and follows at once from #11*07 because 
/(a, /3) is a function of the defining functions of a and /3. 

#20*64. \-:.(a).fa:(a).ga:D.f/3.g/3 
Dem. 

H . #4*2 . (#2007) . D 

h :. (a) ./a : (a) . ga : ^ : (<j>) .f{z (<■& ! *)) : (<£) . ^{S (<£ ! *)} = 

[#10*14] D :/{2 (yfr \z)}.g {2(yjr !*)}:. D h . Prop 

Observe that "/3" is merely an abbreviation for any symbol of the form 
£ (i|r ! z). This is why nothing further is required in the above proof. 

The above proposition is the analogue of #10*14. Like that proposition, 
it requires, for the significance of the conclusion, that / and g should be 
functions which take arguments of the same type. This is not required for 
the significance of the hypothesis. Hence, though the above proposition is 
true whenever it is significant, it is not true whenever its hypothesis is 
significant. 

#20*7. H:(asr):/a.= B .$r!a [#20112] 
This is the analogue of #12*1. 

#20*701. V : ( a <7) :/ [z (<f> ! z), x}.=^ x .g ! [z (</> ! *), x\ 

[The proof proceeds as in #20112, using #1211 instead of #12*1.] 



198 MATHEMATICAL LOGIC [PART I 

#20 702. r : ( a <jr) :/{x, H (<j> I z)} . ~^ x . g ! {*, 2 (<£ ! *)} 

[Proofasin*20'701.] 
#20 703. r : (gp) :/{$ (<j> ! *), a (* ! z)} . =*,* . g ! {2 (<£ ! z), 2 (yfr I z)} 

Dent. 
\-.*lO-3ll.D\-:.f{ x l%dl2}.= Xt6 .gl{ x l%dl1z}:Dz 

<j>lx= xX lx. + lx= x eix.f{ X l%d\2}.= x>9 . 

^\x= xX \x.^\x= x e\x.g\{ X \%d\^) (1) 
h.(l).*ll-n-3-341.D 

h:.Hp(l). ^:(^ x ,e).^\x^ xX \x.f\x= x d\x.f{ X n,d\^}.^^. 
{^ x ,e).<i>\x= xX \x.^\x= x e\x.g\{ X \%d\^}: 

[*20-l.*10-35]D:/{a(</)!^),^(t^)}-=*^..9J{^!^^^} (2) 

h.(2).*10-ll-281.D 

Yi.{^g):f{ X \%e\z].^ 6 .g\{ X \%e\^}:^i 

(W) -f{* (<t> ! •*)• * (* ■ *)} ■ =** ■ 9 ■ {* (0 ■ *)» * (* ■ *)} (3) 

K (3). #1211. DK Prop 

*20'701'702-703 give the analogues, for classes, of #12*11. 

#2071. h:.a = /3. = :gla.D g .glj3 [#2019] 

This is the analogue of #13 01, 

This completes the proof that all propositions hitherto given apply to 
classes as well as to individuals. Precisely similar reasoning extends this result 
to classes of classes, classes of classes of classes, etc. 

From the above propositions it appears that, although expressions such as 
z (<f)z) have no meaning in isolation, yet those of their formal properties with 
which we have been hitherto concerned are the same as the corresponding 
properties of symbols which have a meaning in isolation. Hence nothing in 
the apparatus hitherto introduced requires us to determine whether a given 
symbol stands for a class or not, unless the symbol occurs in a way in which 
only a class can occur significantly. This is an important result, which enables 
us to give much greater generality to our propositions than would otherwise 
be possible. 

The two following propositions (#20-8'81) are consequences of #133. The 
"type" of any object x will be defined in #63 as the class of terms either 
identical with x or not identical with x. We may define the "type of the 
arguments to <f>z" as the class of arguments x for which "<f>x" is significant, 
i.e. the class &(</>&• v ~ </>#). Then the first of the following propositions shows 
that if "(f)(i" is significant, the type of the arguments to <j& is the type of a; 
the second proposition shows that, if "<£a" and "yjra" are both significant, 
the type of the arguments to <££ is the same as the type of the arguments to 
yjrz, because each is the type of a. #20*8 will be used in #63'11, which is a 
fundamental proposition in the theory of relative types. 



SECTION C] GENERAL THEORY OP CLASSES 199 

*20 - 8. h : <f>av<>-><f>a . D . & (<f>xv~<f>x) — ob{x = a.y/.x^a) 

Bern. 

K*13-3.*10-11-21.D 

I- :: Hp. D :.(f)x v~<£#. = x :x = a.v .x^a:. 

[*20'15] D :. £ (<f>xv~<f>x) = x~(x=a .v .x^a)::D\- . Prop 

*2081. H :</>av~^>a.T^av~i/ra.D.^(^>a?v~</>a?) = ^(i|ra;v~'^ra;) 

l-.*20'8.D!-:Hp.D.^(^v~^)=^(aj = a.v.a;4=a) (1) 

I- . *20*8 . D h : Hp . D . $ (yfrxv~y}rx) = cb (x = a . v . x =)= a) (2) 

h . (1) . (2) . *10\L2M3 . Comp . D 

I- : Hp . D.£(<j>xv~<fix) = $(x = a.v.x^a).$(ylrxV'>>ylrx) = $(x=a.v.x^a). 
[*20-24]D . &(</>#v~</>#) ==£(-^r v~i/r#) : D H . Prop 

In the third line of the above proof, the use of #10121 depends upon the 
fact that the "a" in both (1) and (2) must be such as to render the hypothesis 
significant, i.e. such as to render 

significant. Hence the "a" in (1) and the "a" in (2) must be of the same 
type, by *10"121, and hence by *1013 we can assert the product of (1) and 
(2), identifying the two "as." 

Since a type is the range of significance of a function, if <f>x is a function 
which is always true, z (<f>z) must be a type. For if a function is always true, 
the arguments for which it is true are the same as the arguments for which 
it is significant; hence z (<j>z) is the range of significance of $x, if (x) . <f>x holds. 
Thus any class a is a t}^pe if (x) . x e a. It follows that, whatever function <f> 
may be, x (cj>x v ~ <£#) is a ty pe ; and in particular, fc(x = a . v .x^a)is& type. 
Since a is a member of this class, this class is the type to which a belongs. 
In virtue of #20 - 8, if <f>a is significant, the type to which a belongs is the class 
of arguments for which <f>x is significant, i.e. oH {<$>x v ~ (f>x). And if there is any 
argument a for which <fca and yfra are both significant, then <\>ot and yfra: have 
the same range of significance, in virtue of *20'S1. 



*21. GENERAL THEORY OF RELATIONS 

Summary q/"#2I. 

The definitions and propositions of this number are exactly analogous to 
those of *20, from which they differ by being concerned with functions of two 
variables instead of one. A relation, as we shall use the word, will be under- 
stood in extension: it may be regarded as the class of couples (x, y) for which 
some given function ty (x, y) is true. Its relation to the function ^ (&, {/) is 
just like that of the class to its determining function. We put 
*21*01. f{M}1r (x, y)}. = : ( a $) : ! (*, y) . = x<y . + (x, y ) :/{<£ \ (fi f $)} Df 
Here "$yyjr(x, y)" has no meaning in isolation, but only in certain of its uses. 
In #21-01 the alphabetical order of u and v corresponds to the typographical 
order of lb and $ in f{x§y]r (x, y)}, so that 

f{§H(x>y)}- = :(R<f>):<t>l(x,y).= x , v .ylr(x,y):f{<l>l(!) > ti)} Df 
This is important in relation to the substitution-convention below. 
It will be shown that 

i.e. that two relations, as above defined, are identical when, and only when, 
they are satisfied by the same pair of arguments. 

For substitution in <f> ! (oc, §) and cf> ! ($, Sb), we adopt the convention that 
when a function (as opposed to its values) is represented in a form involving 
x and {/, or any other two letters of the alphabet, the value of this function 
for the arguments a and b is to be found by substituting a for £ and b for #, 
while the value for the arguments b and a is to be found by substituting b 
for £ and a for y. That is, the argument mentioned first is to be substituted 
for the letter which comes first in the alphabet, and the argument mentioned 
second for the later letter; thus the mode of substitution depends upon the 
alphabetical order of the letters which have circumflexes and the typographical 
order of the other letters. 

The above convention as to order is presupposed in the following definition, 
where a is the first argument mentioned and b the second: 
#21-02. a {<f> ! (&, §)) &. = .</> ! (a, b) Df 
Hence, following the convention, 

b {<j> ! (x, £)} a . = . <£ ! (b, a) Df 

a{4>l($,x)}b. = .<}>l(b,a) Df 

b {tf> ! (£, tb)}a. = .<f>l (a, b) Df 

This definition is not used as it stands, but is introduced for the sake of 

a {tyf {x, y)}b. = : ( a 0) : ! (x, y) . = x> y .^(x } y): <f>l (a, b) 



SECTION C] GENERAL THEORY OP RELATIONS 201 

which results from #21 01 02. We shall use capital Latin letters to represent 
variable expressions of the form &y<\> ! (x, y), just as we used Greek letters for 
variable expressions of the form-£(<£ ! z). If a capital Latin letter, say R, is 
used as an apparent variable, it is supposed that the R which occurs in the 
form "(R)" or "(giZ)" is to be replaced by "(<f>)" or "(a<£)," while the R which 
occurs later is to be replaced by "£$<£ ! {x, y)." In fact we put 

(22)./R. = .(*)./{^!(* f y)} Df. 
The use of single letters for such expressions as xy<f>(x,y) is a practically 
indispensable convenience. 

The following is the definition of the class of relations : 

#2103. Rel = 5{(a0).i2 = ^!(a?,y)} Df 

Similar remarks apply to it as to the definition of "Cls" (#20*03). 

In virtue of the definitions #210102 and the convention as to capital 
Latin letters, the notation t( xRy" will mean "x has the relation R to y." This 
notation is practically convenient, and will, after the preliminaries, wholly 
replace the cumbrous notation x {scp<f>(x, y)} y. 

The proofs of the propositions of this number are usually omitted, since 
they are exactly analogous to those of #20, merely substituting #1211 for 
#12-1, and propositions in #11 for propositions in #10. 

The propositions of this number, like those of #20, fall into three sections. 
Those of the second section are seldom referred to. Those of the third section, 
extending to relations the formal properties hitherto assumed or proved for 
individuals and functions, are not explicitly referred to in the sequel, but are 
constantly relevant, namely whenever a proposition which has been assumed 
or proved for individuals and functions is applied to relations. The principal 
propositions of the first section are the following. 

#2115. h i. + ix, y) . =*,„.*(*, y) : = . x$^(x, y) = $$ X (x, y) 

I.e. two relations are identical when, and only when, their defining functions 
are formally equivalent. 

#21-31. h :. xfiyjr (x, y) = xy X {oo,y). = '.x {xyyjr (x, y)} y.= x>y .x {xy X (a?, y)} y 
I.e. two relations are identical when, and only when, they hold between 
the same pairs of terms. The same fact is expressed by the following 
proposition: 

#21-43. \-:.R = S. = :xRy.= x>y .xSy 

#212 21 22 show that identity of relations is reflexive, symmetrical and 
transitive. 

#21-3. r- : x {xy-+ (x, y)}y. = .+ (x, y) 

I.e. two terms have a given relation when, and only when, they satisfy its 
defining function. 



202 MATHEMATICAL LOGIC [PART I 

*21151. V .(>&$). &§y\r(x,y) = x$<j>\{x,y) 

I.e. every relation can be denned by a predicative function. Hence when, 
using #21 "07 or #21*071, we have a relation as apparent variable, and are there- 
fore confined to predicative defining functions, there is no loss of generality. 

*2i-oi. /{^^(^y)}.«:(a0)":0K^y)-^ir-^(^y)=/{0K^«)} Df 

On the convention as to order in #21 01 '02, cf. p. 200, and thus relate u, v 
to &, § so that 

fm^(x > y)}. = :(' K <f>): ( f>l(x,y).= x>v .ylr(x,y):f{<f>l(v,^} Df 
#21-02. a{<f>l(x, £)}&. = .<£! (a, b) Df 

#21-03. B,e\ = B{(>a<l>).R = $g<f>l(x, y)} Df 

The following definitions merely extend to relations, with as little modifi- 
cation as possible, the definitions already given for other symbols. 
#21-07. (R) ./R . = .(<]>) .f{x§<j>l(x, y)} Df 

#21071. (a«) ./R . = . (atf>) ./{xp<t> l(x, y)} Df 
#21072. [(iR) (<f>R)] .f{iR) (<f>R) . = : (a#) : <\>R . = R . R = 8 :/S Df 

#21-08. /{££> (R, 8)} . = : (3<£) : f (R, S) .= R , s .<j>l (R, 8) :/{<j> ! (R, H Df 
#21-081. P {<j> I (R, S)}Q. = .<f>l (P, Q) Df 

The convention as to typographic and alphabetic order is here retained. 

#21082. f{R(fR)} . = : ( a <£) :fR.= R .<j>lR :/(f ! R) Df 

#21 083. R e <f> ! R . = .<£! R Df 

#21-1. h :./{xpf (x, y)}. = : ( a 0) : <f> I (x, y) .=,,„. * (x, y) :/{</> ! (fi, v)} 

[*4-2.(*21-01)] 
#21-11. r :. ir (x, y) .=*,„. X (x, y) : D :f{%H (x, y)} . = ./(££% (*. V)) 
[#4-86-36. #10-281. #211] 
This proposition proves that every proposition about a relation expresses 
an extensional property of the determining function. 
#21111. b :.f{<j>l(x,y)} . ^.g{<f>l(x,y)} : D :/{*#$ l(<c, y)\ . =* .g{%$<j>\(x,y)} 

[Fact . #11-11-3 . #10-281 . #211] 
#21-112. h :. (^) :./{££</> ! (x, y)}.=*.gl [%H ! (a, y)} [*121 . #21-111] 

It is #12-1, not #1211, which is required in this proposition, because we 
are concerned with a function (/) of one variable, namely (f>, although that 
one variable is itself a function of two variables. 

#2112. h :. ( a </>) :. <£ ! (x, y) .= x>y . ^ (x, y) \f[M}+ (<c, y)} . s ./{££</> ! (x, y)} 
[#2111. #1211] 
This is the first use of the primitive proposition #12-11, except in 
*20701-702-703. 

#21-13. \-:.y}r(x,y).= Xt y.x(^y)--^- $®t 0> y) = *§X 0> V) 
[#211. #12-11. #13-195] 



SECTION C] GENERAL THEORY OP RELATIONS 203 

*2114. r :. £$f (x, y) = $$ x (*• V) • D : ^ 0> 2/) ■ =*,j/ • % 0*» 2/) 
[Proof as in #2014] 

#2115. h:.^(*,y).s (M ,.x(*.y):s-^^(*»y)-^X( fl? »y) [*2H314] 

This proposition states that two double functions determine the same 
relation when, and only when, they are formally equivalent, i.e. are satisfied 
by the same pairs of arguments. This is a fundamental property of relations 
as defined above (#21-01). 

#21151. V . (a<^>) . $§f (x, y) = m ! (x,y) [#2115 . #1211] 

#2116. l-:(a*):/{^(*,y)}.s./{^!(*,y)} [#2112] 
#2117. h :(<}>). fm<l>l(x,y)}.1-fl$H(x,y)} [#2116. #10-1] 
#2118. h :. a$4> (x, y) - £y> {x, y) . D :/{£y> (*, y)} . = ./ {£y> (*, y)} 

[#211115] 
#2119. V :. £y> (*, y) = £y % («, y) . = : (/) :/! £y> (*, 2/) ■ => -/J ^% (*. y) 
[#2118 . #1011-21 . #211 . #10-35 . (#1301) . #21112 . #10301] 

*2ii9i. v -.. a^r (x, y) = $p x (*. y) • = = (/) -f 1 - *Q * (*, y) ■ = •/'• %vx (*. y) 

[#211819] 
#21-2. r . &y> {x, y) = ££<£ (a;, y) [#21-15 . #4 2] 

#21-21. r : £y> (a?, y) = &y> (a?, y) . = .%^,y)=££<£(#,y) [#21-15 . *10'32] 
#21-22. h: £>£<£ (a?, y) = £y>(#, y) . £y> (*> y) = ^X (*, y) • => ■ 

%<^(^y) = ^x(^y) [#2115. #10-301] 
#21-23. r : ^y <£ (x, y) = £y> (a?, y) . £y </> (ar, y) = ££ X (#, y) . D . 

%f(^y) = ^ X (^,y) [#21-21-22] 

#21-24. H : £y^ (a, y) = £y </> («, y) . M)x ( x > y) = $00 (*> y) • D ■ 

&9lr(w,y) = &9x(*'y) [*21-21-22] 
#21-3. h:x{xyyjr(x,y)}y. = .^(x,y) [#21102 .*10-4335 .#121 1]_ 

This shows that x has to y the relation determined by i/r when, and only 
when, x and y satisfy yfr (x, y). 

Note that the primitive proposition #12-11 is again required here. 

#21-31. r :. M)^{x, y) = ££%(*, y) . = :oc {$pyjr(x,y)} y . =,,„ . x [$$ X ( X >V)) V 

[#21-15-3] 
#21-32. h . £y [# {&y^ (#, y)} y] = &y> (#, y) [#21-315] 

#21-33. Vz.R = ^<f>{x > y).^:xRy.= Xt y.<\>{x,y) [#21-31-3] 

Here R is written for some expression of the form xfity (x, y). The use 
of a single capital letter for a relation is convenient whenever the determining 
function is irrelevant. 

#21-4. H : R e Rel . = . (g<£) .R = $g<f>l (x, y) [#203 . (#2103)] 
#21-41. r . $$<t> (x, y) e Rel [*21-4'15.1] 

#21-42. h.$P(xRy) = R - [#21-315] 

#2143. \-:.R = S. = :xRy.= x , y .xSy [*21'15-3] 

#20-5*51'52 have no analogues in the theory of relations. 



204 MATHEMATICAL LOGIC [PART I 

*21-53. b:.S=R.D s .(f>S: = .(t>R [*101 .*21*2-18*21 . Coram .*10'ir21] 
*21'54. h:.(^S).S = R.<f>S. = .<l>R [#21*18. #10*1 1*23 .#21*2 .#10*24] 
*21*55. I- . $y<j> (x, y) = (1R) [xRy .= x , y .<j> (cc, y)) [*21*33*54 . #141] 
#21*56. h . E ! (iR) {xRy .= x , y .cf> {x, y)} [#21*55 . #14*21] 

#21*57. r :. $$<!> (x, y) = {iR)(fR) .D:g [%§<f> (x, y)j . = . g {(iR) (fR)} 

[#14*1. #21*54. #13*183] 
#21*58. Vixy<j>(x,y) = (iR){R = $cy$(x,y)} [#4*2 .#10-11 .*2154 . #14*1] 

The following propositions are the analogues of #206 ff., and have a similar 
purpose. 

#21 6. h : ( a J?) m fR. = m ~ {(R) . ~/R] [Proof as in #20*6] 
#21*61. h : (R) .fR .D.fS [Proof as in #20*61] 

#21 '62. When/i? is true, whatever possible argument of the form ctycj) ! (x, y) 
R may be, (R) .fR is true. [Proof as in #20-62] 

#21-63. \-:.(R).p v/R .Dzp.v. (R) .fR [Proof as in #20-63] 
#21-631. If "fR" is significant, then if # is of the same type as R, "fS" is 
significant, and vice versa. [Proof as in #20*631] 

#21 632. If, for some R, there is a proposition fR, then there is a function 
fR, and vice versa. [Proof as in #20*632] 

#21*633. "Whatever possible relation R may be, f(R, S) is true whatever 
possible relation S may be" implies "whatever possible relation S may be, 
f{R, S) is true whatever possible relation R may be." 

[Proof as in #20*633] 
#21*64. \-:.(R).fR:(R).gR:D.fS.gS [Proof as in #20*64] 
#21*7. h : (g#) :/R . = R . g ! R - [Proof as in #20*7] 

#21701. \-:(Rg),:f(R,x).= B , x .gl(R,x) [Proof as in #20*701] 
#21*702. h : fag) :f(x, R) . = BjX .gl(R, x) [Proof as in #20*702] 
#21*703. r : (Rg)':f(R, S).= R>s .gl (R, S) [Proof as in #20*703] 
#21*704. I- : (g#) :f(R, a) . = Rt a . g ! (R, a) [Proof as in #20*703] 
#21-705. h : ( a #) :/(a, R).= a , A .g ! (a, R) [Proof as in #20-703] 
#21*71. h:.R = S. = :g\R.D g .glS [Proof as in #20*71] 

From the above propositions it appears that relations, like classes, have 
all the formal properties which they would have if they were symbols having 
a meaning in isolation. Hence unless a symbol occurs in a way in which only 
a relation can occur significantly, we do not need to decide whether it stands 
for a relation or not. This result, like the corresponding result for classes 
mentioned at the end of #20, is important as giving greater generality to our 
propositions than they would otherwise possess. The results obtained in #20 
and #21 for classes and relations whose members or terms are neither classes 
nor relations can be extended, by mere repetition of the proofs, to classes of 
classes, classes of relations, relations of classes, relations of relations, and so on. 



*22. CALCULUS OF CLASSES 

Summary o/*22. 

In this number we reach what was historically the starting-point of 
symbolic logic. The Greek letters used (except </>, yfr, %, 0) are always to 
stand for expressions of the form cb (<f> I x), or, where the Greek letters are 
not apparent variables, ik (<j>x). The small Latin letters may either be such as 
have a meaning in isolation, or may represent classes or relations; this is 
possible in virtue of the notes at the ends of #20 and *21. We put : 

*22-01. aCl3. = :xea.D x .xe@ Df 

This defines " the class a is contained in the class /3," or " all a's are /S's." 
#22-02. ar\P=*&(xea.xe&) Df 

This defines the logical product or common part of two classes a and /3. 
#2203. avj/3 = $i(a:ea.v.xel3) Df 

This defines the logical sum of two classes ; it is the class consisting of all 
the members of one together with all the members of the other. 

#2204. -a = x(x~ € a) Df 

This defines the negation of a class. It is read "not-a." It does not 
contain every object x concerning which " x e a " is not true, but only those 
objects concerning which " xea" is false ; i.e. it excludes those objects for 
which " x e a " is meaningless. Thus it consists of all objects, df the type next 
below a, which are not members of a ; but it does not contain objects of any 
other type but this. 

#2205. a-/3 = an~j3 Df 

This definition gives an abbreviation which is often convenient. 

The postulates required for the algebra of logic have been enumerated by 
Huntington*. In our notation, they are as follows. 

We assume a class K, with two rules of combination, namely u and n ; 
and we then require the following ten postulates : 

la. a v b is in the class whenever a and b are in -the class. 

16. a n b is in the class whenever a and b are in the class. 

II a. There is an element A such that a u A = a for every element a. 

II 6. There is an element V such that a r\ V = a for every element a. 

Ill a. avb = bva whenever a,b, avb and b u a are in the class. 

Ill b. ar\b = b r\a whenever a, b, ar\b and b n a are in the class. 

* Trans. Amer. Math. Soc. Vol. 5, July 1904, p. 292. 



206 MATHEMATICAL LOGIC [PART I 

IV a. a\j(br\c) = (avb)r\(av c) whenever a, b, c, a u b, a u c, b n c, a u (b r\ c), 

and (av b) r\(au c) are in the class. 
IV b. a c\ (b u c) = (a r\b) \j (a r\ c) whenever a, b, c, a r\ b, a r\ c, b u c, an (6 u c), 
and (anb)yj (a n c) are in the class. 
V. If the elements A and V in postulates II a and 116 exist and are 
unique, then for every element a there is an element — a such that 
a\j — a = V and ar\ —a = A. 
VI. There are at least two elements, x and y, in the class, such that x =)= y. 
The form of the above postulates is such that they are mutually inde- 
pendent, i.e. any nine of them are satisfied by interpretations of the symbols 
which do not satisfy the remaining one. 

For our purposes, " K " must be replaced by " Cls." A and V will be the 
null-class and the universal class, which are defined in *24. Then the above 
ten postulates are proved below, as follows : 
I a, in *22-37, namely " h . a u /3 e Cls " 
I b, in *22-36, namely " h . a n /3 e Cls " 
II a, in #24*24, namely "KowA = a" 
II b, in *24-26, namely " h . a n V = a " 
III a, in *22-57, namely "Kowj3 = j8ua" 

III b, in *22-51, namely "h .ar\ fi = fi n a" 

IV a, in #22-69, namely " h . (a u yS) n (a w y) = a w (/3 n 7) " 
IV b, in #22-68, namely " h . (o o /3) u (a n 7) = a n (^ u 7) " 

V, in #24-21-22, namely "h.an-o = A" and "r. aw _a = V" 

VI, in #24-1, namely " r . A + V " 

Hence, assuming Huntington's analysis of the postulates for the formal 
algebra of logic, the propositions proved in what follows suffice to establish 
that this algebra holds for classes. The corresponding propositions of #23 
and #25 prove that it holds for relations", substituting Rel, o, r\, A, V for 
Cls, u, n, A, V. 

The principal propositions of the present number are the following : 

(1) Those embodying the formal rules: 

#2251. Karȣ = y8na 

#22-57. h-.au/8rs0ua 

These embody the commutative law. 

#22-52. h . (a n /3) n 7 = a r\ (/3 n 7) 
#22-7. h . (a u /3) u 7 = a u (/? u 7) 
These embody the associative law. 

#22*5. h . a r\ a = a 
*2256. h . a u a = a 

These embody the law of tautology. 



SECTION C] CALCULUS OF CLASSES 207 

#2268. h . (a n £) u (a n 7) = a n (# u 7) 

#22 69. K(«u j 8)n(au 7 ) = ou(^n 7 ) 

These embody the distributive law. It will be seen that the second 
results from the first by everywhere interchanging the signs of addition and 
multiplication. 
#22 8. I- . - (- a) = a 

This is the principle of double negation. 
#22 81. h:aC/3. = .-/3C-a 

This is the principle of transposition. 

(2) Other useful propositions : 
#2244. h:aC/3./3C 7 .D.aC7 
*22441. h : a C ft. xea.D.xeft 

These embody the two forms of the syllogism in Barbara. 
#22 62. \-:aCft. = .auft = ft 
#22621. h:aC/3. = .an/3 = a 

These two propositions enable us to transform any inclusion (a C ft) into 
an equation. 

#22-91. h.au/3 = au(/3-«) 

I.e. " a or ft " is identical with " a or the part of ft which is excluded 
from a." 



*22-01. aCft. = :xea.D x .xeft Df 
#2202. anft = x(xea.xeft) Df 
#22 03. auft = x(xea.v.xeft) Df 
#2204. -a --=x(x~ea) Df 

#2205. a-ft =an~ft Df 

#221. \-:.aCft. = :x € a.D x .xeft [*4-2 . (#22-01)] 
#22-2. I- . a n ft = % (x € a . x e ft) [*20'2 . (*22'02)] 

#22-3. y-.avft = x(xea.v.xeft) [#20*2 . (#22*03)] 
#22-31. K-« = ^(^ e o) [#20*2 . (#22-04)] 

#22-32. h.a-ft = $(xea.x~€ft) [#20*2 . (#22-05) . *22'2 . *20"32] 

#22-33. \-:x€ar\ft. = .xea.xeft [#203 . *22'2] 

#22-34. \-:.xea\jft. = :xea.v.wep [#203 . *22'3] 
#22-35. \-:xe-a. = .x~ea [#203 . #22*31] 

#22-351. K-a=f=a 
Dem. 

h.*22-35.*5-19.Dt-:~{a;e-a. = .a;ea}: 
[#10-11] Db:(x):~{xe-a. = .xea}: 

[#10251] 0\-:~{(x):xe-a. = .xea}: 

[*20-43.Transp] Dh:~(-o = a):3h. Prop 



208 MATHEMATICAL LOGIC [PART I 

This proposition is used in proving that the null-class is not identical 
with the class containing everything (#24 - l), which is used to show that at 
least two classes exist. Our axioms do not suffice to prove that more than 
one individual exists, but they prove the existence of at least two classes and 
at least two relations. 

#22*36. Kan/3eCls [#20-41] 
#22-37. KauygeCls [#2041] 
#22-38. h . - a e Cls [#20-41] 
#22-39. h . z (<f>z) n % (yftz) = z(<f>z. yfrz) 

Dem. 

h . #22-33 . D h : x e z (<f>z) n 2 {y\rz) .=.xez (<f>z) .xez (yfrz) . 

[#20-3] = .<f>x.^x (1) 

h . (1) . #20-33 . D h . Prop 
#22-391. h .z{<f>z)\j^ (yjrz) = / z{<t>zv ^z) [Similar proof] 
#22-392. \-.-z (<f>z) = z(~<f)z) [Similar proof] 

*22'4. H :.aC/3./3Ca. =:xea. = x . x eft 

Dem. 

h.*22-l . D b :: aC ft . ~ : x e a . D x . x e ft :. ft C a . = : x e ft . D x . x e a :. 

[#4 - 38] D b :: a C ft . ft C a . = :. x e a . D x . x e ft : x e ft . "D x . x e a :. 

[#10-22] =:.xea.= x .X€ft::D\-.Frop 

#22-41. b:aCft.ftCa. = .a = ft [#22'4 . #20-43] 
#22-42. h.aCa [Id. #10-11] 

#22-43. H:a^/3Ca [#3-26 .#1011] 

#22-44. r-:aC/3./3C 7 .D.aC 7 [*10%3] 

This is one form of the syllogism in Barbara. Another form is the following : 
#22-441. bzaCft.xea.D.xeft [#101 . Imp] 
#22-45. t- : a C £ . a C 7 . = . a C£ n 7 

Dem. 

b . #22"1 .Db:.aCft.aCy. = :xea.D x .xeft:xea.D x .x€y: 
[#10*29] =:xea.D x .xeft.X€<y: 

[*22-33.*10-413] =:xea.D x .X€&ny:.Dh. Prop 

#22-46. bzxea.aCft.D.xeft [#22-441 . Perm] 
#22-47. l-:aC 7 .D. anftCy [2243;44] 

#22-48. h-.aCft.D.anyCftny [#10-31] 
#22-481. \-:a = ft.D.any = ftny 

Dem. 

b . #22-41 . D :. Hp . D : a Cft . ft C a : 

[#22*48] DianyCftny.ftnyCany: 

[#22-41] D:an7 = /3n 7 :.DKProp 



SECTION C] CALCULUS OF CLASSES 209 

*2249. \-:aC0.yC8.D.anyC/3n8 [*1039] 

$22*5. h . a n a = a. 

Dem. 

V . $22*33 . D h :. x e a c\ a . = : x e a . x e a : 

[*4-24] = :#ea (1) 

h . (1) . *10'11 . *2043 . D H . Prop 

The above is the law of tautology for the logical multiplication of classes. 

*22 51. Kan£ = /3na [*22'33 . *43 . *1011 . *20'43] 

*22 52. h . (a n /3) n 7 = n (0 n 7) [*2233 . *4'32 . *1011 . *2(V43] 

Thus logical multiplication of classes obeys the commutative and associative 
laws. References to *2233-34-35 and to *2043 will in. future often be omitted. 
*22 53. onj3n 7 = (ani3)ft7 Df 

This definition serves merely for the avoidance of brackets. 
*22-54. h:.a = /3.D:aC7. = .y9C 7 [*20'18] 
*22-55. \-z.a = /3.D:yCa. = .y.C/3 [*2018] 
*22551.\-;a = /3.D.auy = /3uy [*10-411] 
*22 56. Kaua = « [*4-25 .*10'11] 

The above is the law of tautology for the logical addition of classes. 
*22 57. Kau£=£ua [*4\31 . *10-11] 

*2258. \-.aCav/3./3Ca\jj3 [*l'3.*2-2] 

*2259. h:aCy.0Cy. = . a yjj3Cy 
Dem. 

K*221 .D h :z Hy . = :. x e a . D x . x e y : x e fi . % . x e y 1. 
[*10-22] ^z.(x}z.X€a.D.xeyzxe/3.D.x€yz. 

[*4*77.*10-271] = :. (x) :. x e a . v . x e/3 : D . x ey :. 
[*22-34.*10-413] =:.(x):x6ayj/3.D.xey::3\-. Prop" 
The analogue of *4*78, i.e. 

aCfi.v.aCy. = .aCl3yjy 
is false. We have only 

aC0.v.uQy:D.aCfi\jy. 

A similar remark applies to the analogue of *4'79. Cf. *22'64'65. 
*22*6. \-:.xea\J@. = :aCy.@Cy.D y .x€y 

Dem. 
h .*22-59.DH:.aC7.y3C7.D:a;6auy8.D.a;e7:. 
[Comm] Dhz.xeav fi.DzaCy.ftCy. D .xey:. 

[*10-1121] D h :. x e a u /3 . D : a C 7 . /3 C 7 . D v . x e 7 (1) 

h . *101 . D h :. 0C7 . C7 . D y . x e 7 : D : aC a u /S . £ C a u £ . D . # e a v, £ : 
[*2258] D:xea\j/3 (2) 

r . (1) . (2) . D H . Prop 

R&W I 14 



210 MATHEMATICAL LOGIC [PART I 

*22-61. \-:ctCj3.D.aCl3uy [#22-44-58] 
#2262. h:aCj8. = ,auj8 = j8 
Bern. 

Y . #472 . D Y ;: x e oc . D . x e ft : = :. x e a .v . x e @ : = . x e ft :. 

[*22-34] =i.x<-a\j/3. = .a;e/3 (1) 

f-.(l).*10-271.Dr:.: «C/3. = i.xea v £ . -^.ae/S :. 

[#20-43] = :.au£ = £::DI-.Prop 

#22 621. h:aC/3. = .an^ = « .[#4-71] 

The proof proceeds as in #22*62. The proposition #22*621 is one of the 
most useful propositions in the present number. 

#2263. h:au(an J 8) = a [#4*44] 

The process of obtaining #22*63 from #4*44 is of the same kind as the 
process employed in the proofs that have been written out in this number. 
Hence only #4*44 is referred to. We shall similarly restrict references for 
later propositions in this number. The process is always roughly as follows : 
p, q, r are replaced by sea, xefi, xey; then #10-11 is applied, and such 
further propositions of #10 as may be required, together with #2233 34*35. 

*22'631. Kfln(au/3) = o [#22*58*621] 

#22*632. h:a = /S.D.a = ar. / S [#22*42*621] 

#22 633. H:aC i 8.D.au7 = (an/3)u 7 [#22*551*621] 

#22*64. h:.aC7.v./3C 7 :D.arv/3C7 

Dem. 

h . *22*47*51 . D h ; a C 7 . D . a n /3 C 7: /3 C 7 . D . a n /3 C 7 (1) 

K (1). #4*77. DK Prop 

The converse of this proposition does not hold, because the converse of 
#10*41 does not hold. 

#22 65. h:.aCy3.v.aC 7 :D.aC/3u7 [*22*61*57 .#477] 

Here again the converse is untrue. 
#22-66. h:aC i 8.D.«u 7 C i 8u 7 [#238] 

#2268. h ^ (a n 0) u (a n 7) = a n (£ v 7) 

Dem. 

f- .#22*34. D V :: xe [(« n ^) u (a r\ 7)} . = :. ocean @. v .xeOLr><y\. 

[#22*33] = :.xea.xe/3.v.xea.xey:. 

[#4*4] = :. xea: xe (3 .v ,xe<y :. 

[#22*34] = :.xea.xe/3 U7 :. 

[#22-33] =:.xean(/3vry) (1) 

r- . (1) . #1011 . #2043 . D r- . Prop 



SECTION C] CALCULUS OF CLASSES 211 

#22*69. K(«uj3)ft(au7) = au(j3ft7) [Similar proof, by #4 -41] 

The above propositions #22-68"69 are the two forms of the distributive 
law. Note that either results from the other by interchanging the signs of 
addition and multiplication. 

*227. K(auj3)u 7 = ou(i3u7) [#4-33] 

#22 71. «uj3u7=(ou^)u7 Df 
#2272. h:aC 7 ./3CS.D.au/3C 7 uS [*3'48] 
#2273. h:a = 7 .j8 = S.D.av,/3 = 7 uS [#10-411] 
#2274. h:on i 8C7.aft7C i 8. = .ani3 = aft7 

Dem. 
b . #22*43 . #473 . Db:ar\/3Cy. = .an/3Ca.an/3Cy. 
[#2245] =.an|9Can7 (1) 

h .(l)X' /3 - Dh:an 7 C/3. = .an 7 Can / 6f (2) 

P> 7 

h . (1) . (2) . *4-38 .DI-:an/SC 7 .an 7 C/3. = .an/3Can 7 .ar. 7 Can y g. 

[#22-41] = .an£ = an 7 :Dh. Prop 

#228. h.-(-a) = a [#413] 

#22-81. H:aC/3. = .-/3C-a [#4"1] 

*22811. h:aC-^.E.^C-a [#41.*22-8] 

#22-82. h:anySC 7 . = .a- 7 C-/3 [#414] 

#22-83. H:a = /3. = .-a = -/3 [#4-11] 

#22-831. l-:o = -/3.E./3 = -a [#412] 

#22-84. K-(an/3) = -av-/3 [#4-51] 

#22-85. Kan£ = -(-av-£) [#22-84-831] 

#22-86. K-(-an-£) = au/8 [#4-57] 

#22-87. h.-aft-j8 = -(«u|8) [*22'86831] 

*22-8485-86-87 are De Morgan's formulae. 
#22-88. h. (a?), are (aw -a) [#2-11] 

This is a form of the law of excluded middle. 
#22-89. b . (a?) . *~ e (a - a) [*3'24] 

This is a form of the law of contradiction. 
#229. K(av/3)-/3 = a-/3 [#5-61] 

#22-91. r-.au/3 = au(£-a) 

Dem. 

b . #5 - 63 . DH:.#ea.v.#e/S:E:#ea.v .xe/3 . ar~ea :. 
[*22-33-34-35] Dh :. areau/3.s:area. v. are (£-<*): 
[#22-34] =:ar€Ou(/9-o) (1) 

h . (1) . #10-11 . #20-43 .Db. Prop 

14—2 



212 MATHEMATICAL LOGIC [PART I 

*2292. H:aC£.D./3 = av(/3-a) [#22'91*62] 
*22-93. r-.a-/9 = a-(an/3) 
Dem. 

I- . #4*73 . Transp . D I- :.#ea. D :#~e/3.= .^(aiea.xe ft). 

[*22-33] E.^e^n^):. 

[#5'32] D I- ;.3cea.x~el3 . = .xea.x~e(an ft) :. 

[*22'35-33] :>r:#ea-£. = .a e a-(an/3): 

[*10-11.*2043] Dh.a-/3 = a-(an/9).Dh.Prop 
*22'94. r:(a)./a.= .(a)./(-a) 

Dem. 

K*10\L. Dh:Ya)./a.D./(-a): 

[*1 0-11-21] Dh:(a)./«.D.(a)./(-a) (1) 

h . *101 . D h : (a) ./(- a) . D ./{_ (- «)} . 

[*22'8.*20\L8] 3-/«: 

[♦10-11-21] Dh: (a) ./(-a). >.(«). /a (2) 

h . (1) . (2) . D h . Prop 

This proposition is used in connection with mathematical induction, in 
*90-102, which is required for the proof of *90132, which is one of the 
fundamental propositions in the theory of mathematical induction. 

*22-95. H:(aa)./a.s.( a a)./(-a) 

Dem. 

K*22-94.Dh:(a).~/a. = .(a).~/(-a) (1) 

H . (1) . Transp . *20'6 . D I- . Prop 



#23. CALCULUS OF RELATIONS 
Summary of #23. 

The definitions and propositions of this number are to be exact analogues 
of those of #22. Properties of relations which have no analogues for classes 
will not be dealt with till Section D. Proofs will be omitted in the present 
number, as they are precisely analogous to those of analogous propositions in 
#22. In this number, as always in future, capital Latin letters stand for 
expressions of the form ob§<f> ! (x, y), or, where they are not being used as 
apparent variables, for &$<£(#, y). The principal propositions of this number 
are the analogues of those of #22. 



#2301. RG.S. = zxRy.3 x>y .xSy Df 
#2302. Rf>S = $§(xRy.xSy) Df 

#2303. RvS = ob§ (xRy . v . xSy) Df 
#23-04. ±R = $p{~ (xRy)} Df 

#2305. R^-S=Rn^S Df 

Similar remarks apply to these definitions as to those of #22. 
#23-1. h :. R Q S . = : xRy . D x>y . xSy 
#23-2. \-.RnS = x-$ {xRy . xSy) 
#23-3. \-.RuS = x~§ (xRy . v . xSy) 
#23-31. h . -^ R = ty { co (xRy)\ 
#23-32. I- . R-^S= x§ [xRy . ~(xSy)} 
#23-33. \-:x(RnS)y. = .xRy.x8y 
#23-34. f- :. x (R \j 8) y . = : xRy .v.xSy 
#23 35. h : x-^Ry . = .~(xRy) 
#23-351. h.-R^R 
#23-36. KiZntfeRel 
#23-37. KiZotfeRel 
#23-38. h.^-ReUel 

#2339. h . $$4> {x, y) n $$+ (x, y) - £$ {£ (*, y) . + (x, y)} 
#23-391. r . x$<f> (x, y) c; xpyjr (x, y) = $$ {<f> (x,y).v.yfr (x, y)} 
#23-392. V . - x$<f> (x, y) = $$ { ~ $ (x, y)} 
#23-4. V \.R QS.SQR. = :xRy. =*,„ . xSy 
#23-41. \-:RGS.SGR.= .R = S 
#23-42. b.RGR 
#23-43. \-.RnSdR 
#23-44. \-'.RGS.SGT.D.RGT 
#23441. b-.RQS.xRy.O.xSy 



214 MATHEMATICAL LOGIC [PART I 

*2345. biRQS.RQT.D.RQSnT 

*23'46. H : xRy .RdS.D.xSy 

*2347. \-:RGT.D.R*SGT 

*2348. \-:RGS.D.RnTGSnT 

*23481. ^:R = S.D.RnT=SnT 

*2349. \-:PGQ.RGS.D.PnRGQr*S 

*23-5. \-.RnR = R 

*2351. \-.RnS = SnR 

*23 52. K(EnS)nT=En(/SnT) 

*2353. RnSnT=(RnS)nT Df 

*2354. h:.P = £.D:PGr. = .#GT 

*2355. h:.# = £.:>:rGP.==.TG£ 

*23551. h:P = S.:>.PiyT=#vyr 

*23 56. \-.RvR = R 

*23 57. \-.RvS = SvR 

*2358. V.RZRvS.SGRvS 

*23'59. \-:R<iT.S(iT. = .RvSGT 

*236. \-:.x(RvS)y.= :RGT.S<iT.3 T .xTy 

*23 61. hiPGS.D.PGSc/T 

*23 62. H:PG,Sf. = ..Rvy,Sf=£ 

*23621. h:PG£. = .Pn£ = £ 

*23'63. h.Rv(RnS) = R 

*23631. h.E/S(Ea>Sf) = E 

*23632. \-:R = S.D.R = RnS 

*23 633. h: J RG*S.D.^c/T = ( J RniSf)oT 

*2364. \-:.RGT.v.SGT:D.RnS<iT 

*2365. hr.EG^.v.EGTzD.i^G^vyr 

*23 66. \-:RdS.D.RvT<iSvT 

*23'68. h.(R*S)v(RnT) = Rn(SvT) 

*2369. h .(Rv S) * (Rv T) = R v (S n T) 

*237. b.(RvS)vT=Rv(SvT) 

*2371. RvSvT=(RvS)vT Df 

*2372. h-.PGE.QGS.D.PvyQGEc/S 

*23 73. h:P = P.Q = >Sf.D.PvyQ = Pvy^ 

*2374. \-:PnQGR.PnR<lQ. = .PnQ = PnR 

*23& h.-(-P) = P 

*2381. hziZGS.EE.-SG^-iZ 

*23811. h:PG-#. = .£G^J2 

*2382. h^nSGr.^.P-TG^S 

*2383. h:P = £. = .-^-E=-£ 



SECTION C] CALCULUS OF RELATIONS 215 

*23 831. h:R = -S. = .S = -R 

*2384. b.-(RnS) = -R^-S 

*23-85. \-.RnS = -(-^Rv-S) 

*2386. \-.^(-R^^8) = RvfS 

*2387. \-.-Rn-L.S = -(RvS) 

*2388. \-.(x,y).oc(Rv-^R)y 

*23 89. \-.{x,y).~\x(R-R)y] 

*23 9. \-.(RvS)-S=R-S 

*23 91. \-.R\jS = Rv(S^-R) 

*2392. b:RGS.D.S = Rv(S-R) 

*23 93. h . R-S = R-(R * #) 

*2394. \-:(R).fR.==.{R).f(-R) 

*23 95. Ma^./tf.EE.^)./^!?) 



*24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE 
EXISTENCE OF CLASSES 

Summary q/"#24. 

The universal class, denoted by V, is the class of all objects of the type 
which, in the given context, is being denoted by small Latin letters, i.e. of 
the lowest type concerned. Thus V, like " Cls," is ambiguous as to type. Its 
definition is as follows : 

*2401. V = &(# = «) Df 

Any other property possessed by everything would do as well as " x — x," 
but this is the only such property which we have hitherto studied. 

The null- class, denoted by A, is the class which has no members. Like 
V, it is ambiguous as to type. We use the same symbol, A, for null-classes 
of various types ; but these null-classes differ. The type of A is determined 
by that of the terms x concerning which "aseA" is false : whatever x may be, 
" xeA." will not represent a true proposition, but unless x is of the appropriate 
type, " x e A" will be meaningless, not false. Thus A is of the type next above 
that of an x concerning which "x e A" is significant and false. The definition 
of A is 

*2402. A = -V Df 

When a class a is not null, so that it has one or more members, it is said 
to exist. (This sense of "existence" must not be confused with that defined 
in *1402.) W T e write " g ! a " for " a exists." The definition is 

*2403. a!a. = .(ga;).#ea Df 

In the present number, we shall deal first with the properties of A and V, 
then with those of existence. In comparing the algebra of symbolic logic with 
ordinary algebra, A takes the place of 0, while V combines the properties of 
1 and of oo . 

Among the more important properties of A and V which are proved in 
this number are the following : 

*241. h.A+V 

I.e. " nothing is not everything." This is useful as giving us the existence 
of at least two classes. If the monistic philosophers were right in maintaining 
that only one individual exists, there would be only two classes, A and V, 
V being (in that case) the class whose only member is the one individual. Our 
primitive propositions do not require the existence of more than one individual. 



SECTION C] THE EXISTENCE OF CLASSES 217 

*24102103 show that any function which is always true determines the 
universal class, and any function which is always false determines the null- 
class. 

*24 2122 give forms of the laws of contradiction and excluded middle, namely 
" nothing is both a and not-a " (a r» — a = A) and " everything is either a or 
not-a "(au-« = V). 

#242324 2627 give the properties of A and V with respect to addition and 
multiplication, namely : multiplication by V and addition of A make no change 
in a class (#24-2624) ; addition of V gives V, and multiplication by A gives A 
(#24-27-23). It will be observed that the properties of A and V. result from 
each other by interchanging addition and multiplication. 

#243. h:aC/9. = .a-/3 = A 

I.e. " a is contained in /S " is equivalent to " nothing is a but not j3" 
*24311. h:«C-/3. = .an/3 = A 

I.e. " no a is a /3 " is equivalent to " nothing is both a and ft." 

*24'411. h:j8C«.D.B = )8u(a-j8) 

#24 43. h:a-/3Cy. = .aC#v 7 

As a rule, propositions concerning V are much less used than the corre- 
lative propositions concerning A. 

The properties of the existence of classes result from those of A, owing to 
the fact that a ! a is the contradictory of a = A, as is proved in #24-54. Thus 
we have, in virtue of #243, 

#2455. h~(aCi8). = .a!a-i3 

I.e. " not all a's are /9's " is equivalent to " there are «'s which are not /3's." 
This is the familiar proposition of formal logic, that the contradictory of the 
universal affirmative is the particular negative. 

We have 
#24 56. h:.a!(ouj8). = :a!o.v.a!j8- 
#24 561. H:a!(an/3).D.g!a.a!/3 

I.e. if a sum exists, then one of the summands exists, and vice versa ; and 
if a product exists, both the factors exist (but not vice versa). 

The proofs of propositions in the present number offer no difficulty. 



#24-01. V = £(a> = a?) Df 

#2402. A = -V Df 

*2403. a !a. = .( a *). # 6 a Df 

#241. b . A + V [*22-351 . (*24'02)] 

*24-101. h . V = - A 022-831 . (*24*02)] 



218 



MATHEMATICAL LOGIC 



[PART I 



*24102. I- : (x) . (f>cc . = . 2 (<f>z) = V 
Dem. 

t- . #1315 . #5501 . D h :. (fyx . = : (fix . = . x = x :. 
[#1011-271] D h :.(x). <j>x . = : (x) : (j>x . = . x = x : 

[*20- 15] = : 2 (<£*) = S (ar = a;) : 

[(#24-01)] = : $ (</>*) = V :. D h . Prop 

Thus any function which is always true determines the universal class, 
and vice versa. 

#24103. h : (x) . ~ <j>x . = . 2 (<f>z) =-- A 
Devi. 

V . #24102 . D h :. <» -~</># ■ = : 2(~£s) = V 
[*22-392] = :-2(^) = V 
[#22-831] =:t(0s) = -V: 
[(#24-02)] = : £ (</>*) = A : . D I- . Prop 

#24104. h.(ar).ajeV 

Dem. 

\-.^20S.D\-:xeY . = .x^x (1) 

1- .(1) . *13'15 . #1011-271 .31-. Prop 
*24105. h .(x).x~eA 
Dem. 

h. #22-35. Dh:a;€A. = .a5~eV: 

[#412] DH:«;~eA. = .tfeV (1) 

K (1) . *1011-271 . *24-104< . D h . Prop 
#2411. K(o).oCV 
Dem. 

h . #24-104 . #101 . DKareV. 
[Simp] "Dh : xea.D .xeY : 

[*10-11.*22-1] Dh:aCV: 

[#10-11] D h : (a) . a C V : D I- . Prop 

#2412. K(a).ACa 
Dem. 

V . #24105 . #10-1 . DK«~eA. 

[#2-21] Dr-:a?eA.D.a?ea (1) 

h . (1) . #1011 , #22-1 . D t- . Prop 

#2413. h:a=A. = .aCA 

Dem. 

V . #2412 . #473 .Dh:aCA. = .aCA.ACa. 
[#22-41] = . a = A : D h . Prop 

#24-14. r- : (a?) . x e a . = . a = V 

h. #24-102. D [-:(*). are a. = J(ajeo) = V. 
[#20-32] = . a = V : D r- . Prop 



SECTION C 



THE EXISTENCE OP CLASSES 



219 



#24141. hVCa. = .V = a 

Bern. 

Y . #2411 . #4-73 .Dh:VCa. = .aCV.VCa. 

[#2241] = . a = V : D h . Prop 

#2415. V ;(x).x~ea.= . a = A 

Bern. 

V . #24103 v Dh:(*).^efl. = J(*ea) = A. 
[#20-32] =.a = A:Dh. Prop 

#2417. r:a = V.=s.-a = A [#22-83 . (#24-02)] 

#24-21. h.an-a = A [#24103 . *22\89] 

#24-22. Kav,-a = V [*22'88 . #24102] 

#24-23. KanA = A [#2412 . #22621] 

#24-24. h.auA = a [#2412 . #2262] 

The above two propositions (#24-23-24) exhibit the algebraical analogy of 
A to zero. 
#24-26. K«nV = « [#22-621. #2411] 

This exhibits the analogy of V to 1. 
#24-27. h.awV = V [#22-62 .#2411] 

This exhibits the analogy of V to oo . 

#24-3. h:aC/3. = .a-/3 = A 

Bern. 

V . #4-53-6 . D 

h : . x e a . D . x e ft : = : ~ (x e a . x<^> e ft) : 

[#22-35] = :~(xea.xe-ft): 

[#22-33] = :~(xea-ft) 

h.(l).*1011-27l.D 

h : a C ft . = . (x) . ~ (x e a — ft) . 

[#2415] =.a-/3 = A.OKProp 
The above proposition is very frequently used. 
#24-31. h:aC/3. = .-av/3 = V 
Bern. 

r- .#4*6 . ~2>\- •..xeoL.'S ,xe ft : = zx^€a..v .xe ft i. 



(1) 



[#1011-271] Dh:. a C/S.= 
[#22-35] = 

[#22-34] = 

[#24-14] = 



(x) : x~ea . v .xe ft : 
{x) zxe — a.v.xeft: 
(x) . x e (— a u ft) : 
-au/3=V:.Dh.Prop 



This proposition is the correlative of #24-3, but, unlike that proposition, 
it is not useful in the sequel. Every proposition concerning A has a corre- 
lative concerning V, but we shall often not give these correlatives, since they 
are seldom required for subsequent proofs. 



220 MATHEMATICAL LOGIC [PART I 

*24311. f-:aC-/3. = .an/3 = A 
Bern. 

h . *22-35 .Dh:.xea.D.X€-l3: = :x€a.D.x^e0: 

[*4"51-62] = :~(xea.xe0): 

[*22-33] =:~(* € ar>/3) (1) 

> . (1) . *10 11271 . D h : a C - /3 . = . (x) . #~e a n . 

[*24-15] = .an/3 = A:Dh.Prop 

*24312. H : -a C/3 . = . a u /3 = V 

h . *22-35 . D H :. - a C /S . = : a?~e a . D, . a? e £ : 

[*4*64] = :(#):#ea.v.tf;e/3: 

[*22:34] = :(x).xeav/3: 

[*24-14] =:aw/3 = V:.DH.Prop 

*24 313. h:an/3 = A. = .a = a-/3 [*24-311 . *22"621] 
*2432. H:.oui9 = A. = .a = A.^ = A 
Dem. 

V . *2413 .Dh:.«u J 8 = A.= :au i 9CA: 

[*22-59] =:aCA./3CA: 

[*24'13] =:a=A./3 = A:.D!-.Prop 

*2433. h:a = V.D.aw/3 = V 

I- . *22551 .Dh:Hp.D.av£ = Vu/3 
[*24-27.*22-57] = V:Dh.Prop 

*24'34. h:a = A.D.an/3 = A [*22-481 . *24-23] 

*2435. H:a = V.D.an/9»/S [*22481 . *24"26] 

*2436. h:o = A.D.au/3 = /9 [*22551 . *2424] 

*24 37. \-:.ar\/3 = A. = :ocea.y€/3.D Xt y.a:^y 
Dem. 

h.*2415 .Dh:.an/9 = A.== : (x) . x~e(a r\ /3) : 
[*22-33] =:(*).~(a;e«.*6/9): 

[*13- 1 91] = :(x,y):x = y.D.~(x€a.ye/3): 

[Transp] = : (x, y) : x e a . y e @ . D . x j= y :. "D \- . Prop 

*2438. h:.an/3 = A.D:a4=yS.v.a = A.y8=A 
Dem. 

h . *22481 .Dh:ar>£ = A.a = /3.D.ar>a = A. 
[*22-5] D.a = A. 

[*20-23] Z>.a = A./3 = A (1) 

h . ( 1 ) . Ex p . D h : . a n /3 = A . D : a = . D . a = A . £ = A : 
[*4-6] Z>:a + /3.v.a = A./3 = A:.DKProp 



SECTION C] THE EXISTENCE OP CLASSES 221 

*2439. H.an/8 = A. = :#ea.D a ..a:~e£ [*24'311 .*2235] 
*244. h:an/3 = A. = .(au/3)-a = /3. = .(au£)-£ = a 

Dem. 

\- . *24-31 l.Dh:an0 = A. = .0C-a. 

[*22 621] s./3-a = /3. 

[*22-9] =.(au / 8)-a = / S (1> 

k(l)^. Dh:/3na = A. = .08u«)-/3 = «: 

[*22-51-57] Dh:an/3 = A. = .(au/3)-/3 = o (2) 

h . (1) . (2) . D h . Prop 

*24401. h : #C a . D . (£ u 7) - a = 7 - a 
Dem. 

K*2268. DI-.(^u 7 )-a = (/3-a)u( 7 -a) (1) 

K*243. DH:Hp.D. / 8-a = A (2) 

h . (1) . (2) . D h : Hp . D . 08 u 7 ) - a = A w ( 7 - a) 
[*2424] = 7 -a:DKProp 

*24402. \-:an0=A..t-Ca.r}C0.D.%r\ v = A 
Dew. 

h . *22'49 . D h : Hp . D . £ n 77 C a n /3 . 
[*22-55] D.^n^CA. 

[*24-13] D.^ V = A:D\-. Prop 

*2441. h . a =* (a n 0) w (a - /S) 
Dem. 

K *22-68. D K (a n £)«(«-£) = an OS u-,8) 
[*24-22] =«nV 

[*24-26] = a.DKProp 

*24411. h:/3Ca.D.a = i 8u(fl- J 8) 
Dem. 

H..*22-633^|^-^.DI-:y8Ca.D./3w(a- / 8) = (an/3}u(a-/3) 
[*24-41] ' ' = a: Dr. Prop 

*24412. h:y8Ca. 7 C/3.D.(a-/8)u(/3- 7 ) = a- 7 
Dem. 

h.*24-41.DH:Hp.D.(a~ / 3)w(y8- 7 ) = (a-^n 7 )w(a-/3- 7 )u( i 8- 7 ) 
[*24-3-23] = (a - - 7) u (0 - 7) 

[*22-68] =={(a-/3)u/3}-7 

[*24-411] =a-7:DI-.Prop 

This proposition is used in *234*181, in the theory of continuous functions. 
*24'42. h:an/8C7.a-/3C7. = .aC7 
Dem. 

r . *22-59 .3\-:an0Cv.a-0Cry.==.(cLr\0)v(a-0)Cy. 
[*24-41] =.oC7:Dh.Prop 



222 MATHEMATICAL LOGIC [PART I 

#2443. h:«-/3C7. = .aC/3w 7 
Dem. 
h . #5'6 . "D h :: x e a . x<^> e /3 . "D . x e y : = :. x e a. D : x e ft . v . % e y :. 
[#22-35*33] D h :: x e a — ft . D . x e y : = :. x e a . D : x e /3 . v . x e y :. 
[*22-34] =.:.««.D.«e(/3u 7 ) (1) 

h.(l).*10-ll-271.Dh.Prop 

*24431. \-.(cHJy)o(/3v-y) = (anj3)u(<x-y)v(/3ny) 

This and the following proposition are lemmas for #24"44. 

Dem. 

V . #22-68 . D V . (a v y) n (/3 u - 7 ) = {(a u 7 ) n £} v, {(a u 7 ) r\ - y] 
[*22-68] =(an/3)u( 7 n/3)vj( a - 7 )u( 7 - 7 ) 

[#24r21] =• (a n /3) v (y n /3) u (a - 7 ) u A 

[*24-24] =(an/3)v,( 7 n£)u(a- 7 ) 

[#22-51-57] = (a n £) u (a - 7) u (£ n 7) . D H . Prop 

#24432. h.(a-7)u(j8n7) = (on i 8)u(a-7)u( J 8n 7 ) 

h . *24-22-35 . D h . a n /3 = (a n /3) n (7 u - 7) 

[#22-68] -(an J 8n 7 )u(an/3- 7 ) 

[#22-51] =(ar\fir\y) u (a n -70 /3) . 

[#22-551] D K (a n /3) u (a — 7) = (a r> /? n 7) v (an — 7 n /3) u (a - 7) 

[#22-63] =(any3r>7)o(a-7) 

[#22-57] =(a-7)u(an/8r»7). 

[#22-551] D h . (a r> /3) u (a - 7) u (/3 n 7) = (a- 7) u (on /3 r> 7) w (/3 n 7) 

[#22-63] = (a - 7) v (£ n 7) . D r- . Prop 

#24-44. I- . (a u 7) n (/3 u - 7) = (a n - 7) u (/3 n 7) [#24-431 -432] 

#24*45. I- : (a n 7) u (/3 - 7) = A . = . /3 C 7 . 7 C - a 

Dem. 

h . #24-32 . D I- : (a n 7) v (/3 - 7) = A . = . a n 7 = A . /3 - 7 = A . 

[#24-3-311] =. 7 C-a>./3C7:Dh.Prop 

#24-46. f- : (a n 7) u (/3 - 7) = A . D . a n /3 = A 

Dem. 

h . #24-45 . #2244 . D H : Hp . D . /3 C - a . 

[#22811] D.aC-/3. 

[#24-311] D.an/3 = A:DKProp 

The following propositions, down to #24'495 inclusive, are lemmas inserted 
for use in much later propositions, most of them being only used a few times. 



SECTION C] THE EXISTENCE OF CLASSES 223 

*2447. !-:ar»/3 = A.au/3 = 7. = .aCy./3 = 7-a 

Dem. 
K*24-311.DI-:an/3=:A. = ./3C-a (1) 

h . *2241 . DF:au,8=7. = .au/3C7.7Cav;/3. 

[*22-59.*24-43] =.aC7.^C 7 . 7 -aCi3 (2) 

K(l).(2)Oh:a«£ = A.au/8 = y.=s.£C-a.aC7. / 8C7.y-aC£. 
[#4'3] =.aCy.^Cy.y8C-a.y-aC/9. 

[#22-45] = .aCy./8Cy-a.y-aC£. 

[#22-41] =.aCy. / S = y-a:Dh.Prop 

#24-48. h:.| : Ca.fCa.77C/3.7/'C/S.an/3 = A.D: 

|wi7 = fwV ■ = ■? = !'. 17 = V 
Dem. 

K #22-73. Dh:f = f .i7 = V.3>.|w^ = f «V (1) 

F . #22-481 . Dh:.£v V = £'v V '.D:(£v v )na = (£'v v ')na: 

[#22-68] D:(|na)w(i 7 na) = (fn«)u(Vna) (2) 

h. #22-621. DH:fCa.D.fna= s f:|'Ca.D.fna=f: 

[#347] Dh:fCa.fCa.D.fna = f.fno = f (3) 

h . #22-48 . Dh:,Cj8.D.j?naConj8: 

[#22-55] Dh:97C/3.an^=A.D.^naCA. 

[#24-13] D.7;na = A (4) 

Similarly h:i/'C^.an/8 = A.D.V«a = A (5) 

I- . (3) . (4) . Dh:.Hp.D:(^a)u( }/ na) = ^A 

[#24-24] = £ (6) 

r . (3) . (5) . D h :. Hp . D : (£' ^ju^aj^'uA 

[#2424] =|' (7) 

H.(2).(6).(7).Dh:.Hp.D:fu7; = f uV.D.^^r (8) 

Similarly f- :. Hp . D : £v 1; = £' w 77' . D -v^v' (9) 

h.(l).(8).(9).Dr.Prop 

The above proposition, besides being used in the next two, is used in the 
theory of couples (*54 - 6), in the theory of greater and less (#117-632), and in 
the chapter on the ordering of classes by the principle of first differences 
(#170-68). 

#24481. r:.an/3 = A.any = A.D:au/3 = auy. = ./3 = y 

Dem. 
h . #24-48 -L~^-- r J^-l . D 

h :.aC a .aC a. ft C - a .yC~ a. a — a = A . D : 

a w/3 = a w 7 . = . a = a ./3 = y (1) 
h . #22-42 . #24-21 . D 

h:.aCa.aCa./3C — K.^C-a.a- a = A . = . /3 C — a^C- a . 



224 MATHEMATICAL LOGIC [PART I 

[*24-311] = .an£ = A.an 7 =A (2) 

K*20'2.*4-73.DH:a = a./?=7. = .£ = 7 (3) 

K(l).(2).(3).Dh.Prop 

The above proposition is used in the theory of selections (#83*74), in the 
theory of greater and less (#ll7 - 582), and in the theory of transfinite induction 
(*257). 

*24482. h:.£Ca.77C£.a«/3 = A.D:£u^ = au/3. = .£=a.77=/3 
#24-48 t^4 - #22-42 ] 

$>y J 

The above proposition is used in the theory of convergence (#232-34). 

#2449. h:.arȣ==A.D:aC/3u 7 . = .aC 7 

Dem. 

b . *22621 .Dl-:oCj9w7. = .o=an(^u 7 ) 

022/68] = (an£)u(an 7 ) (1) 

K #24*24 . Dh:an/3=A.D.(an^)u(an 7 ) = on 7 (2) 

I- . (1) . (2) . D h :. Hp . D : a C £ v 7 . = . a = a n 7 . 

[*22621] = .aC 7 :DKProp 

*24491. hj3n7 = A.«Cj3u7. 

D . a — fi = a.r\ 7 .a — y = an j3 .a = (a — /3) u (a — 7 ) 
2)em. 

K*22'621. DhHp.D.o = an(/3u 7 ). 
[*22-481] D.o-7 = «n(^y 7 )- 7 

024-4] = an/3 (1) 

Similarly h : Hp . D . a-£=ar» 7 (2) 

h.(l).(2). Dh:Hp.D.(a-/3)v(a- 7 ) = (an 7 )u(an£) 

022-68] = a n ( 7 v £) 

022-621] =a (3) 

I- . (1) . (2) . (3) . D b . Prop 
The above proposition is used in the theory of selections (*8363-65) and 
in the theory of segments of a series (#211-84). 

#24-492. h.)3Ca.a-i3 = 7.D.a-7 = j8 

Dem. 

K #22-481 . D h : Hp . D . a- 7 = a- (a-/3) 

O 22 ' 8 ' 86 ] =an(-au£) 

O 22 * 8 ' 9 ] = «'>£ 

2 2-621] =/3:Dh.Prop 

The above proposition is used fairly frequently, especially in the theory of 
series. It is first used in *93'273, in the theory of "generations." 



SECTION C] THE EXISTENCE OF CLASSES 225 

#24493. h:/Sn 7 = A.D.« = («-/9)u(«-7) 

Dem. 

\- . #22-84 . #2417 . 3 h : Hp . 3 . - /3 w - 7 = V . 

[#24*26] 3.a = an(-£u- 7 ) 

[*22'68] = (a-/3)u(a- 7 ):3r .Prop 

#24494. K-fCa.i7C£.an£ = A.D.(f ui7)-a = i7.(£ui7)-£ = f 

K#24-3. 3h:Hp.3.£-a = A (1) 

K#24-31I. 3h:Hp.3./3C-a. 

[#22-44] 3 . v C - a . 

[#22-621]' .3.17-0 = 17 (2) 

K #22-68. 3K(£ui7)-a = (£-a)v(i7-a) (3) 

h.(l).(2).(3).*24-24.3l-:Hp.3.(^ui 7 )-a = i7 (4) 

Similarly h : Hp . 3 .(%v v )-0= £ (5) 

h . (4) . (5) . 3 I- . Prop 

This proposition is used in the theory of selections (#83'63 and #88'45). 

#24-495. h:ar» 7 = A.3.(au 7 )-( / 3u 7 ) = a-/3 

Dem, 

h . #22-87-68 . 3 

H . (a u 7) - (£ v 7) = (a - £ - 7) u (7 - £ - 7) 
[#24-21] = a _ / 3- 7 (1) 

K*24-311.#22-621.3H:Hp.3.a-7=a (2) 

h.(l).(2). 3 K Prop 

The above proposition is used in the theory of minimum points 
(*205-83-832-84). 

In the remainder of this number we shall be concerned with the existence 
of classes. Many of the properties of the existence of classes follow from the 
fact that to say a class exists is equivalent to saying that the class is not equal 
to the null-class. This is proved in #24-54. 

#24-5. b-.ftla. =2. fax). <cea [*42 . (#24-03)] 

#24 51. h : ~g ! a . = . a = A 

Dem. 

I- . #24-5 . 3 I- : ~jj ! a . = . ~ {(g.r) . x e a] . 

[#10-252] =.(«).aj~ea. 

[#24-15] = .a = A:3b.Prop 

#24-52. h . g ! V [#24-51-1 . Transp] 

This proposition states that the class of all objects of the type in question 
is not null, but has at least one member. The assumption that there is some- 
it &w 1 15 



226 MATHEMATICAL LOGIC [PART I 

thing, which is equivalent to this proposition, is implicit in the proposition 
#10*1, that what is true always is true in any instance. This would not hold 
if there were no instances of anything; hence it implies the existence of 
something. It will be observed that the above proposition (#24*52) depends 
on #24*1, which depends on #22*351, which depends on #10*251, which depends 
on #10*24, which depends on #10*1 or on #9*1. The assumption that there is 
something is involved in the use of the real variable, which would otherwise 
be meaningless. This is made explicit in #9*1, and in the proof of #9*2, which 
is the same proposition as #10*1. 

#24*53. K~g!A [#24*51 . #20*2] 

#24-54. h:g!a.= .a=|=A [#24*51 . Transp] 

#24*55. *-:~(aC/3). = .a!a-£ [#24*3 . Transp . #24*54] 

#24*56. h.a!(auj3). = :a!a.v.a!/3 [#10*42 . #22*34] 

#24*561. h : g ! (a n £) . D . a ! a . a ! /3 [#10*5 . #22*33] 

#24*57, l-:.ar>£ = A.D:g;!a.D.a=f£ 
Bern. 

h . #22*481 .DI-:an/3 = A.a-=/S.D.ar»a = A. 

[#22*5] D . a = A . 

[#24*51] 3..~3!a (1) 

h . (1) . Exp . Transp . D h . Prop 

#24*571. H: a !a.a = / 8.D.a!(artyS) 

Dem. 

V . #24 * 5 7 . C omm . D h : . a ! a . D : a n j3 = A . D . a + /3 : 

[Transp] D :« = £. D .ar\j3$ A . 

[#24*54] ' D.g!(an/3) (1) 

K (1) . Imp . D H . Prop 

#24*58. h:.aC/3.0:a!a.D.a!/9 [#10*28] 

#246. l-:.aC/3D:a + /3.= .a'-/3-a 

Dem. 

h . #22*41 . Transp . O h :. Hp . D : a + £ . D .~(/3 C a) . 

[#24*55] D.a!/3-a (1) 

h. #24*21. Oh:a = /3.D./3-a=-A (2) 

K (2) . Transp . #24*54 .31- : a * £ -a.D . a + /3 (3) 
K(l).(3)'. Dh.Prop 

#24*61. H:~ a !£.D.av/8 = a [*24*51*24] 

#24*62. h:~a!£.D.an£ = A [*24'51-23] 



SECTION q] THE EXISTENCE OF CLASSES 227 

#2463. I- :. A~€K . = zae/e.D*. g!a 

In this proposition, the conditions of significance require that k should 
be a class of classes. The condition "a e k . D tt • 3 J a" is one required as 
hypothesis in many propositions. In virtue of the above proposition, this 
hypothesis may be replaced by "A~e/c." 
Bern. 

V . *13191 .Dh:.A~e/c. = :a = A.D a .a~e«: 
[Transp] =:ae/c.D .a4=A: 

[*24-54] s : a e k . D a . 3 ! a :. D h . Prop 

This proposition is frequently used in later par^ts of the work. We often 
have to deal with classes of existent classes, and the most convenient form in 
which to state that all the members of a class of classes exist is "A~e/c." 



15—2 



#25. THE UNIVERSAL RELATION, THE NULL RELATION, AND 
THE EXISTENCE OF RELATIONS 

Summary q/"#25. 

This number contains the analogues, for relations, of the definitions and 
propositions of #24. Proofs will not be given, as they proceed precisely as 
in #24. 

The universal relation, denoted by V, is the relation which holds between 
any two terms whatever of the appropriate types, whatever these may be in 
the given context. The null relation, A, is the relation which does not hold 
between any pair of terms whatever, its type being fixed by the types of the 
terms concerning which the denial that it holds is significant. A relation R 
is said to exist when there is at least one pair of terms between which it holds ; 
"R exists" is written "g ! R." 

The propositions of this number are much less often referred to than those 
of #24, but for the sake of uniformity we have given the analogues of all 
propositions in #24, with the same numeration (except for the integral part). 

All the remarks made in #24 apply, mutatis mutandis, in the present 
number. 



#25-01. V = $p(x = x.y = y) Df 

#2502. A = -V Df 

#2503. a ! R . = . (ga?, y) . xRy Df 

#251. KA + V 

#25101. r.V=^-A 

#25-102. r : (x, y) . <j> (x, y) . = . 5$ <f> (x, y) = V 

#25-103. \-:(x,y).~<f> (x, y) > = . &§<j> (x, y) = k 

#25-104. h . (x, y) . xYy 

#25-105. h . (x, y) .~(xAy) 

#2511. \-.(R).RQV 

#2512. \- . (R) . A G R 

#2513. \-:R = A. = .RCA 

#25-14. h:(x,y).xRy. = .R = Y 

#25141. \-:YQR. = .Y = R 

#25-15. I- : (x, y) . ~ (xRy) . = .R = A 

#2517. \-:R = Y. = .-^R = A 

*25-21. \-.Rn^-R = A 



SECTION C] THE EXISTENCE OF RELATIONS 229 

*25-22. \-.Rv-^R = Y 

#2£23. h.RnA = A 

*25-24. KPoA=P 

*25 26. I- . R A V = R 

*2527. KPvyV = t 

*25*3. \-:R<ZS. = .R-^S = A 

*2531. h:PG#. = .-uPiy,Sf = V 

*25-311. \-:RQ^-S.~.RnS = A 

*25312. h:^-PG£. = .Pc/# = t 

*25 313. h:RnS = A. = .R-8 = R 

*2532. K:Pvy£ = A.2.P = A.# = A 

*2533. h:P = V.D.Pc/#=V 

*2534. H:P = A.:>.Pn,Sf=A 

*25 35. l-:P = V.D.Pn£ = £ 

*2536. \-:R = A.D.RvS = S 

*2537. H :: R r\ S = A . = \.xRy . zSw . ^ x , y , z , w :oc^z.v .y^w 

*2538. h.EA^=A.D:E^.v.ii = A.-Sf = A 

*25 39. b:.Rr\S = A.= : xRy . D x>y ,~(a:Sy) 

*254. h:Pn(3 = A.= .(PoQ)-P=Q. = .(Pc;Q)^Q = P 

*25401. \-:QGP.D.(QvR)-P = R-P 

*25402. \-:PnQ = A.RQP.SdQ.D.RnS = A 

*2541. \-.R = (RnS)v(R^S) 

*25411. !-:>8'G J R.D.i2 = 5c/(iB^^) 

*25-412. h:QGP.£GQ.D.(P-Q)vy(Q^S) = P-,S 

*2542. h:PnQGR.P-^Q(iR. = .PG.R 

*25'43. \-:P^.QGR. = .P(lQvR 

*25'431. h.(PvyJS)n(Qvy^i2) = (PnQ)vy(P- J R)a(QnE) 

*25432. \-.(P^R)v(QnR)=(PnQ)v(P^.R)v(QnR) 

*25'44. I- . (P vy 22) A (Qv-R) = (P fy^-R) c/(QnP) 

*25-45. h : (P n P) o (Q-P) = A . = . Q G P . P G^P 

*25 46. h:(PniJ)a(Q^) = A.D.PnQ = A 

*25 47. h:PAQ = A.PiyQ = P. = .PGP.Q=P^-P 

*2548. h :: K GP . P' GP . £ G Q . £' G Q . P n Q = A . D : 

Pc/£=P / c;,Sf , .= .P = P'.,Sf=,Sf' 
*25'481. f-:.PnQ = A.PnP = A.D:Pe/Q = Pe/P. = .Q = P 
*25 482. h:.RGP.SGQ.PnQ = A.D:RvS = PvQ.==.R = P.S=Q 
*2549. h:.PnQ = A.D:PGQvyP. = .PGP 



230 MATHEMATICAL LOGIC [PART I 

♦25-491. \-:Q*R = A.PGQvR.3. 

P^Q = PnR.P±R = PnQ.p = (P±Q)v(P^ R ) 
*25-492. \-:QGP.P^Q = R.D.P^R = Q 
*25493. f-:QnP = A.D.P = (P^Q)ei(P-^P) 

*25494. t".RGP.SGQ.P*Q-A.^ m (RvS)±P-S.(RvS)±Q = R 
*25495. H:PAP = AO.(Pc/P)^(^c/P) = P-i.Q 
*25 5. f- : g ! R . = . ( a #, y ) . ^ 

*25'51. H:~g!P. = .P = A 

*2552. Kg!V 

*2553. K~g!A 

*25'54. h:g!P. = .P^A 

*25 55. h~(iJGS). = .a!iJ^ 

*2556. h:.g!(Bvy£). = :a!E.v.a!£ 

*25561. h:a!(JJnS.).D.a!iJ.a!^ 

*2557. f-:.Pn-Sf = A.D:g!P.D.P4=^ 

*25-571.. I- : a ! R . R = £ . D . a ! (R n S) 

*2558. H.PGS.DrgliZ.D.alS 

*25-6. \-:.RGS.D:R$S. = .rIS-R 

*2561. h :~a ! £ . D . P vy £= # 

*2562. h:~a!£.:>.Pn,Sf = A 

*2563. l-:.A~6«. = :22eff. Dje.gl.fi. 



SECTION D 

LOGIC OF RELATIONS 

Id the present section we shall be concerned with such of the general 
properties of relations as have no analogues in the theory of classes. The 
notations introduced in this section will be used constantly throughout the 
rest of the work, and the ideas expressed in the definitions will be found to 
be of fundamental importance. 



*30. DESCRIPTIVE FUNCTIONS 

Summary o/*30. 

The functions hitherto considered, with the exception of a few particular 
functions such as a n /3, have been propositional, i.e. have had propositions for 
their values. But the ordinary functions of mathematics, such as x*, sirxx, 
log as, are not propositional. Functions of this kind always mean "the term 
having such and such a relation to x." For this reason they may be called 
descriptive functions, because they describe a certain term by means of its 
relation to their argument. Thus " sin 7r/2 " describes the number 1 ; yet 
propositions in which sin tt/2 occurs are not the same as they would be 
if 1 were substituted for sin w/2. This appears e.g. from the proposition 
" sin tt/2 = 1," which conveys valuable information, whereas " 1 = 1 " is trivial. 
Descriptive functions, like descriptions in general, have no meaning by them- 
selves, but only as constituents of propositions*. 

The general definition of a descriptive function is : 
#30-01. R'y = (ix)(xRy) Df 

That is, " R'y " is to mean " the term x which has the relation R to y." 
If there are several terms or none having the relation R to y, all propositions 
about R ( y, i.e. all propositions of the form "<f>(R'y)," will be false. The 
apostrophe in "R'y" may be read "of." Thus if R is the relation of father 
to son, " R'y " means " the father of y." If R is the relation of son to father, 
"R'y" means "the son of y"; in this case, all propositions of the form 
" $ (R'y) " will be false unless y has one son and no more. 

All the functions that occur in ordinary mathematics are instances of the 
above definition ; all are obtained in the above manner from some relation. 
Thus in our notation "R'y" takes the place of what would commonly be 
"fy" tn * s latter notation being reserved for propositional functions. We 
should write "sin 'y" in place of "siny," using "sin" to' express the relation 
of x to y when x = sin y. 

A definition such as R'y = (ix)(xRy), where the meaning given to the 
term defined is a description, must be understood to mean that the term 
defined (in this case R'y) and the description assigned as its meaning (in this 
case (ix) (xRy)) are to be interchangeable in use : the definition is, in a sense, 
more purely symbolic than other definitions, since the description assigned as 
the meaning has itself no meaning except in use. It would perhaps be more 
formally correct to write 

f(R'y). = .f{(ix)(xRy)) Df. 
* Cf. *14, above. 



SECTION D] DESCRIPTIVE FUNCTIONS 233 

But even this definition would not be quite complete, because it omits 
mention of the scope of the two descriptions. R'y and (7a?) (xRy). Thus the 
complete form would be 

[R'y-].J(R'y). = .\(ix){xR y )-\.f{{ix)(xRy)) Df. 
But it is unnecessary to adopt this form of definition, provided it is under- 
stood that the definition #30-01 means that "R'y" may be written for 
" (ix) (xRy) " everywhere, i.e. in indications of scope as well as elsewhere. The 
use of the definition occurs always in accordance with the proposition : 

r : [R'y] .f(R'y) . =s . [(ix) (xRy)] .f(ix)(xRy), 
which is #301, below. 

It is to be observed that #30*01 does not necessarily involve 

R'y = (ix)(xRy). 
For this, by the definition, is equivalent to 

(ix) (xRy) = (ix) (xRy), 
which, by #14-28, only holds when El(ix)(xRy),i.e. when there is one term, 
and no more, which has the relation R to y. 

All the conventions as to scope explained in #14 are to be transferred to 
R'x, i.e., in the absence of any contrary indication, the scope of R'x is to be 
the smallest proposition, enclosed in dots or other brackets, in which the R'x 
in question occurs. 

We put 
#3002. R t S'y=R'(S'y) Df 

This definition serves merely for the avoidance of brackets. It is to be in- 
terpreted as meaning 

[R<S'y].f(R'S<v). = .[R<(S<y)-].f{R<(S<y)} Df. 
In future, we shall often define a new expression as having a descriptive phrase 
for its meaning ; in such a case, the definition is always to be interpreted as 
above. That is, any proposition in which the new expression occurs is to be 
the proposition which is obtained by substituting the old expression for the 
new one wherever the latter occurs. 

R'(S'y), in the above, is to be interpreted by first treating S'y as if it 
were not a descriptive symbol, and applying #3001 and #1401 or #1402 to 
R'(S'y), and by then applying #30*01 and #1401 or #14*02 to S'y. 

The majority of the propositions of the present number are immediate 
consequences of the corresponding propositions in #14. Thus #14-31 — - 34 and 
#14*113 lead immediately to #3012 — *16, which show that, either always or 
when R'y exists, the "' scope " of R'y or of R'y and S'y makes no difference 
to the truth- values of such propositions as we are concerned with. We have 
#30-18. h :, E ! R'y : (z) . <f>z : Z> . <f> (R'y) 



234 MATHEMATICAL LOGIC [PART I 

so that what holds of everything holds of R'y, provided R'y exists. This 
results immediately from #1418, and shows that, provided R'y exists, the fact 
that " R'y " is an incomplete symbol does not prevent its being substituted 
as a value of z whenever we have (z) . (f>z, or an assertion of the propositional 
function <f>z. 

One of the most used propositions of this number is : 

#303. h:.x=R'y. = :zRy.=~ z .z = x 

which results immediately from #14202. The following analogous proposition 
results from the above by means of #14122 : 

#3031. h :. x = R'y . = : xRy : zRy .D z .z = x 

I.e. "x = R'y" involves, in addition to " xRy," the statement that what- 
ever has the relation R to y is identical with x. 

A proposition constantly referred to is : 
#3037. b:ElR'y.y = z.D.R'y = R'z 

In the hypothesis, E ! R'y might be replaced by E ! R l z, but one or other 
of them is essential. For, by #1421, " R'y = R'z " implies E ! R'y and E ! R'z 
(these are equivalent when y = z), and therefore cannot be true when R'y and 
R'z do not exist. 

The use of #3037 is chiefly in cases where y or z or both are replaced by 
descriptive functions. Suppose, for example, that z is replaced by S'w. By 
#3018, we may substitute S'w for z if S'w exists. By #14*21, both sides of 
the implication in #30*37 will become false if S'w does not exist, and there- 
fore the implication will still hold. Hence whether S'iv exists or not, we may 
substitute it for z and obtain 

h : E ! R'y . y = S'w .D.R'y = R'S'w. 
In like manner, if we replace y by T'v, we obtain 

V : E ! R'T'v . T'v = S'w . D . R'T'v = R'S'w. 

A very important proposition is : 
#304. b:.RlR'y.D:a = R'y. = .aRy 

This proposition states that, provided R'y exists, to say that a is the term 
which has the relation R to y is equivalent to saying that a has the relation 
R to y. Thus for example " a is the occupier of the house y " is equivalent 
to " a occupies the house y," " a is the writer of Waverley " is equivalent to 
" a wrote Waverley," " a is the father of y " is equivalent to " a begot y." But 
we cannot argue from " John Smith inhabits London " to " John Smith is the 
inhabitant of London." 

We shall introduce in this and subsequent sections many constant relations 
for which E ! R'y is always true. When R is such that E ! R'y is always true, 

we have, in virtue of #30*4. 

a — R'y . = . aRy 



SECTION D] DESCRIPTIVE FUNCTIONS 235 

for every possible value of y. The following proposition is useful in cases where 
both R and S are such that R'y and S' y always exist : 
#3041. h\.(y).R'y = S'y.= :(y).ElR'yzR = S 

. Thus if we know that R'y and S'y are always identical, we know not only 
that R and S are identical, but also that R'y (and therefore S'y) always exists. 



#3001. R'y = (ix)(xRy) Df 
#3002. R'S'y = R'(S'y) Df 

In interpreting R'(S'y), S'y is to be treated as an ordinary symbol until 
R l (S'y) has been eliminated by #3001 and #1401 or #1402, and then the 
above definitions are to be applied to S'y. 

#301. h : [R'y] ./(R'y) -s . [(ix) (xRy)]./(ix) {xRy) [*4-2 . (#30-01)] 
#3011. h :. [R'y] ./(R'y) . = z (a&) : xRy .= x .x=bz/b [#301 . #141] 

The following propositions are immediate applications of #14*31 ff., made 
in accordance with #30'1. 
#3012. \-zzElR'y.3z.[R'y],pv X (R'y). = zp.v.[R'y]. X (R'y) 

[*14'31] 
#3013. \- zzEl R'y. Dz.[R'y].~ x (R'y). = .~{[R'y]. X (R'y)}[*U-m 
#30-14. b :: E ! R'y . D :. [R'y] .pl X {R'y) • = -P ■ 3 ■ {R'y] . % (12'y) 

[#14-33] 
#30-141. h :: E ! R'y . D :. [fi*y] . x (2ty) D^ . = : [ity] . X (R'y) . D . p 

[#14-331] 
#30142. h :: E ! R'y . D :. [R'y] .p= X (R'y) . = :p.= . [R'y] . X (R'y) 

[#14-332] 
#30-15. \-z.pz [R'y] . X (R'y) z = : [R'y] .p. X (R'y) [*14'34] 

The following two propositions are immediate consequences of #14-113'112. 
#3016. b:[R'y]./(R'y,S'z). = .[S'z]./(R%S'z)[*U-113] 
#3017. \-:.[R'y]./(R'y,S'z). = z 

(g;6, c) : xRy .= x .x = bz xSz .= x .x = c :/(b, c) [#14-112] 
#3018. Yz.E\R'y\(z).$z:1.4>(R'y) [#14-18] 

#3019. h:.R'y = b.D:yjr(R'y). = .y}rb [#1415] 

#30-2. fz. E ! R'y . = : ( a &) : xRy .= x .x^b [*4-2 . #14-11 . (#3001)] 

In proving #30*2, we have to use the definition #3001, not #30'1, because 
E ! (ix) (<f>x) is not of the form /(ix) ($x). This appears if we attempt to apply 
the definition #14'01 to E ! (ix) (<f>x), which leads to an expression containing 
the meaningless constituent E ! b. But by the definition #3001, every typo- 
graphical occurrence of the symbol "R'y" means what results when this 
symbol is replaced by " (ix) (xRy)," hence " E ! R'y " means " E ! (ix) (xRy)" 



236 MATHEMATICAL LOGIC [PART I 

#30-21. H : : E ! R'y . = :. ( a a>) . xRy : xRy . zRy .D x>z .x = z 
[#14203 . (#3001)] 

#3022. f- : E ! R'y . = . R'y *= (ix) {xRy) [*14'28 . (#30-01)] 
Note that we do not necessarily have 

R'y = (ix)(xRy% 
which is only true when E ! R'y. 

#303. h:.x = R'y. = :zRy.= z .z = x [#14-202] 

#30 31. h :. x = R'y . = : xRy :zRy .D z .z = x [#14-122 . #30-3] 

#3032. b : El R'y. = . (R'y) Ry [#14'22] 

#3033. H :: E ! R'y . D :.y(R'y) : = : (ga?) . «.% . f# : = : xRy .D x .fx 
[#1426] 

#3034. f- :. «£y .= x .xSy: D : E ! i2'y . = . E I S'y [#14-271] 

*30341. I- :. xRy . = x . xSy : D : E ! R'y . = . E'y = S'y 

I- . #14-21 . 3 k: R'y = S'y . D . E ! E'y (1) 

f- . #1427 . Comm ,D I- :. Hp . I) : E ! £'y . O . £'y = #'2/ ( 2 ) 

K(l).(2). Dh.Prop 

#3035. b :. R = S. 0:EI R'y. = . El S'y [*30'34 . #21-43] 

#3036. h : E ! R'y . R = S . D . R'y = S'y [#14-27 . Imp . *21'43] 

#3037. \-:ElR'y.y = z.D.R'y=R'z 

Bern. 

h. #14-28. D h :El R'y. 3. R'y = R'y (1) 

(-. #1312. Dh,:.y = z.D:R'y = R'y. = .R'y = R'z (2) 

h.(l). (2). Ass. Dh.Prop 

This proposition is .very frequently used. 

#30-4. h:.ElR'y.D:a = R'y. = .aRy [#14-241] 

This is a very important proposition, of which the use is constant. 

#30-41. h :. (y) . R'y~ S'y . = : (y) . E -! i^'y : R = S 

Dew. 
I- . #14-21 . #1011-27 . D h : (y) . R'y = S'y. O. (y) . El R'y (1) 

I- . #1 4-13-1 42 . D t- :. (y).R'y = S'y.O:(x,y):x = R'y. = .x = S'y : 

f(l).#30-4] D : (x, y) : #% . = . xSy : 

[#21-43] D:R=S (2) 

h . #30-36 . D h : Ea 22'y . R = S.D.R'y^S'y: 

[#1011-27-35] D h :. (y) . E ! R'y: R = S:0. (y) . R'y = S'y (3) 

h.(l).(2).(3). Dh.Prop 



SECTION D] 



DESCRIPTIVE FUNCTIONS 



237 



#3042. \-:.(y).KlR'y.O:(y).R'y = S'y. = .R = S [#3041] 

The hypothesis (y) .El R'y is fulfilled by a number of important special 
relations, of which examples will occur in the subsequent numbers of the 
present section. 

#305. I- : E ! P'Q'z . D . EX Q'z 
Dem. 



K #302 . DH.E! P'Q'z. = 
[#101] D 

[*13-15] D 

[*14'21] D 

*30501. I- :<f).(P'Q'z) . = . ( a 6, c) .c=Q'z .b = P'c . cf>b 

On the meaning of " <j> (P'Q'z)" see note to the definition #30*02. 
Dem. 



{^b)zxP(Q'z).^ x .x = b: 
(>&b):bP(Q'z). = .b = b: 
(< a b).-bP(Q't): 
El Q'z:. Dh.Prop 



:.(g&) : bP (Q'z) : xP (Q'z) . y x . x = b : <j>b :. 
: : . (g&) : . (gc) : c= Q'z : &Pc : #Pc . D^ . x = 6 : <£& : . 
; :."(a&, c) . c = Q'z . 6 = P'e. $& :: D I- . Prop 

#30 51. I- : b = P'Q'.z . = . ( 3 c) . b = P*e . c = Q'z [#30-501 . *13'195] 
#3052. 1- : E ! P'Q'z . = . ( a 6, c) . 6 = P'c . c = Q'z [#30-51 . #14204] 



H . #14-1 122 . D I- ::xf> (P'Q'z) . = : 

[#14-205] 

[#14T22'202] 



*31. CONVERSES OF RELATIONS 

Summary o/#31. 

If R is a relation, the relation which y has to oc when xRy is called the 
converse of R. Thus greater is the converse of less, before of after, husband of 
wife. The converse of identity is identity, and the converse of diversity is 

diversity. The converse of R is written R (read "P-converse"). When 

,\j . • 

R = R, R is called a symmetrical relation, otherwise it is called not-symmetrical. 

<*/ 

When R is incompatible with R, R is called asymmetrical. Thus "cousin" is 
symmetrical, "brother" is not-symmetrical (because when x is the brother of 
y, y may be either the brother or the sister of x), and "husband" is asym- 
metrical. 

The relation of R to R is called "Cnv." It will be shown that every 
relation has one, and only one, converse; hence, applying the notation of #30, 

that one is Cnv'P. Thus R = Cnv'P. We have thus two notations for the 
converse of R; the second is more convenient for the converse of a relation 
not denoted by a single letter. 

The more important propositions of the present number are the following: 

#3113. h.E!Cnv'P 

I.e. any relation P has a converse. Hence the relation "Cnv" verifies the 
hypothesis (y) . E ! R'y, i.e. we have (P) . E ! Cnv'P. 

#3132. r:P = Q. = .P = Q 

I.e. two relations are identical when, and ODly when, their converses are 
identical. 

#3133. H . Cnv'Cnv'P = P 

I.e. any relation is the converse of its converse. 

Very many of the subsequent uses of the notion of the converse of a 

relation require only the propositions which embody the definitions of P and 
Cnv, namely 

#3111. h : xPy . = . yPx 

and 

#31-131. \-;x{Cnv'P)y. = .yPx 



SECTION D] CONVERSES OF RELATIONS 239 

#3101. Cnv = §P{xQy.= x>y .yPx), Df 

#3102. P = %$(yPx) Df 

#311. \-:.QCnvP.E=:xQy.= XtV .yPx [#213 . (#3101)] 

#31101. \-:QCnvP.RCnvP.D.Q = R 

Dem. 

V . *31'1 . D H : . Hp . D : xQy . = XtV . yPx : xRy .= XiV . yPx : 

[*ll-37l] D : xQy . = x>y . xRy : 

[*21'43] D : Q = R : . D h . Prop 

*31*11. \-:xPy. = .yPx [*21"3 . (#31*02)] 

#31111. h.PCnvP [#31111] 

#3112. KP = Cnv'P 
Dem. 

K #31101. Dr:QCnvP.PCnvP.D.Q = P: 

[#31-111] Dh:QCnvP.D.Q = P (1) 

K(l). #1011. #31111. D 

H:PCnvP:gCnvP.DQ.Q=P: 
[#30-31] Dh.P = Cnv'P 
#3113. KE! Cnv'P [#14-21. #3112] 

#31131. h:x(Cnv'P)y.=i.yPx [#3111-12 .#21-43] 

#31132. h : Q Cnv P . = . Q = Cnv'P . = . Q = P [*304 . *311312] 
#3114. KCnv'(PnQ) = Cnv'Pr»Cnv'Q 

I- . #31-131 . D H : # (Cnv'(P nQ)}y . = .y(P nQ)x. 

[#21-33] =.yPx.yQx. 

[#31-131] ~. x (Cnv< P)y.x (Cnv' Q)y. 

[#21-33] = . x {Cnv'P n Cnv'Q} y (1) 

h . (1) . #1111 . *21-43 . D I- . Prop 

#3115. K Cnv'(Pc/#) = Cnv'P c/Cnv'Q [Similar proof] 

#3116. KCnv'^-P = -i- (Cnv'P) 

Dem. 

H . #31131 . D f- : #(Cnv<-^P)y . = . y^-Px . 
[#23-35] = .~(yPa:). 

[#31-131] = . ~ {# (Cnv'P) y) . 

[*23'35] =. a{^_ (Cnv'P)} y (1) 

r . (1) . *1111 . *21-43 . D h . Prop 



240 MATHEMATICAL LOGIC [PART I 

*3ri7. b :. y = P'x . = : xPz . = z . * = y [*30"3 . #31-11] 

#3118. b'..KlP<x. = :(<&y)ixPz.= z .z = y [*30"2 . #31-11] 

#31-21. KCnv'A = A 

Dew. 

h . #31-131 .Db:x (Cnv'A) y. = . yAx : 

[#25-105] Db.~x(Cw'A)y (1) 

I- . (1) . #11-11 . #25-15 . D b . Prop 

#31-22. h . Cnv'V = V [Similar proof] 

#3123. h:P = V. = .P = V 

Demi. 

b.*25U.Db:P = V . = .(x,y).xPy. 
[#31-11 .#1 1-33] = . (x, y) . yPx . 

[#11-2] =.(y,x). yPx . 

[#25-14] = . P = V : D h . Prop 

#31-24: b:P = A. = .P = A [Similar proof] 

#31-32. h:P = Q. = .P = Q 

Dem. 

b . #21-43 .Db:.P = Q. = : xPy .= x<y . xQy : 

[#4-86-21. #31 11] = : yPx . = t , v - vQ®'- 

[*ll-2] = : yPx -= y ,x- yQ% ■ 

[#21-43] = : P = Q : . D K Prop 

*31 ; 33. KCav'Cn v'P = P 

Dem. 

b . #31-131 .Db:x (Cnv'Cnv'P)y . = . y (Cnv'P) a: . 

[#31-131] =.xPy (1) 

1- . (1) . #11-11 . #21-43 . D I- . Prop 

#31-34. r-:P=Q. = .0 = P 
Dem. 

h. #31-32.3 I- :P = Q.= .P = Cnv'Q 
[#31-12-32] = Cnv'Cnv'Q 

[#31-33] =Q:OKProp 

#314. h:PGQ.= .PGQ [#3111 .#11-33] 

#31-41. b:PQQ. = .PGQ [*314-3312] 

#315. h:a!P. = .g[!P [*3124 . Transp .*25'54] 



SECTION D] CONVEBSES OP EELATIONS 241 

#31-51. H:(P)./P. = .(P)./P 

Dem. „ 

h.*101. Dh:(P)./P.D./P: 

[*10 1121] D H : (P) ./P . D . (P) ./P (1) 

K*101.*3ri2.D 

H:(P)./P.D./(Cnv'P). 
[*31'33-12] 3./P: 

[*1011-21] Dh:(P). /P. D.(P)./P (2) 

h.(l).(2).DI-.Prop 

*31'52. ^:(aP)./P.s.( a P)./P [*31'51.Transp] 



e & wi l6 



*32. REFERENTS AND RELATA OF A GIVEN TERM WITH 
RESPECT TO A GIVEN RELATION 

Summary o/#32. 

Given any relation R, the class of terms which have the relation R to a 
given term y are called the referents of y, and the class of terms to which a 

given term # has the relation R are called the relata of x. We shall denote by 

-* 4- ' ' . ' 

R the relation of the class of referents of y to y, and by R the relation of the 

class of relata of x to x. It is convenient also to have a notation for the rela- 

— > 4- — ► 

tions of R and R to R. We shall denote the relation of R to R by "sg," where 

"sg" stands for "sagitta." Similarly we shall denote by "gs" the relation of R 
to R, to suggest an arrow running from right to left instead of from left to right. 

R and R are chiefly useful for the sake of the descriptive functions to which 

— > 4— 

they give rise ; thus R l y = x (xRy) and R'x = § (xRy). Thus e.g. if R is the 

— > 4- 

relation of parent to son, R'y = the parents of y, R'x = the sons of x. If R is 

— ► 
the relation of less to greater among numbers of any kind, R'y = numbers less 

4- —* 

than y, and R'x = flumbers greater than x. When R'y exists, R'y is the class 

whose only member is R'y. But when there are many terms having the 

relation R to y, R'y, which is the class of those terms, supplies a notation 

which cannot be supplied by R'y. And similarly if there are many terms to 

4- 
which x has the relation R, R'x supplies the notation for, these terms. Thus 

for example let R be the relation "sin," i.e. the relation which x has to y when 

4— 

x = sin y. Then "sin'x" represents all values of y such that x — s,my, i.e. all 
values of sin -1 x or arcsin x. Unlike the usual symbol, it is not ambiguous, 
since instead of representing some one of these values, it represents the class 
of them. 

— » 4- 

The definitions of R, R, sg, gs are as follows: 
*3201. JR = ay {« = x (xRy)} Df 
*32-02. *R = /3${/3 = P(wRy)} Df 
*3203. sg = AR(A=R) Df 

*32-04. gs = AR(A=% Df 

In virtue of the above definitions, we shall have sg'.R = R, gs'R — R. This 
gives an alternative notation which is convenient in dealing with a relation 
not represented by a single letter. 



SECTION D] REFERENTS AND RELATA OF A GIVEN TERM 243 

It should be observed that if R is a homogeneous relation (i.e. one in 

which referents and relata are of the same type), then R and R are not 
homogeneous, but relate a class to objects of the type of its members. 

In virtue of the definitions of R and R, we shall have 

#3213. V.~R'y = tc(xRy) 

*32131. \-.R'x = §(xRy) 

—> «- 

Thus by #14*21, we always have E ! R l y and E ! R l x. Thus whatever 

— > «— 

relation R may be, we have (y) . E ! R'y and (#) . E ! R l x. We do not in 

— > «— 

general have (y) . g ! R'y or (x) . g ! R'x. Thus taking R to be the relation 

■-** *— 

of parent .and ehild, R'y = the parents of y and R l x = the children of x. 

<— <— — * — ► 

Thus R'x = A, i.e ~$lR'x, when # is childless, and R'y = A, i.e. ~g ! -R'y, 

— > —> 

when y is Adam or Eve. The two sorts of existence, E ! R'y and g ! R'y, 

can both '-.be significantly predicated of R'y, because "R'y" is a descriptive 

. # «— 

function whose value is a class; and the same applies to R'x. It will be seen 
_> _►, 

that (by #1421) -j ! jft'y . Du E ! R'y, but the converse implication does not 
hold in general. 

We have 

*3fcl6. ±Cr=1$. = .*rJs. = .R = S 

Aso by #3218181, 

V ;x eR'y . = . xRy . — .ye R'x. 

Thus by the use of R'y or R'x, every statement of the form "xRy" can 
be reduced to a statement asserting membership of a class. Since, however, 
the class in question is given by a descriptive function, and descriptive 
functions are defined by means of. relations, we do not thus obtain a method 
of reducing the theory of relations to the theory of classes. 



#32 01. R = &p {a •= x (xRy)} Df 
#3202. *R = /3x{/3 = y(xRy)} Df 
#32 03. sg = AR(A = R) Df 

#3204. •ga-=.AR(A~=m) Df 

— > 
#321. 1- : aRy .= .a=$ (xRy) [*21-3 . (#32-01)] 

#32101. I- : /3Rx . = ./3 = y (xRy) [*21"3 . (#3202)] 

— > 
#3211. h . £ (xRy) —R'y [#321 . #30'3] 



16—2 



244 



MATHEMATICAL LOGIC 



[PART I 



#32111. h . § (xRy)= R'x [*32'101 . #30*3] 
#3212. KElflty [#32-11. #14*21] 

#32121. I- . E ! R'x [#32111 . #1421] 

—* — -> 

" E ! R'y " must not be confounded with " g ! R'y." The former means 

that there is such a class as R'y, which, as we have just seen, is always true; 

— > 
the latter means that R'y is not null, which is only true if y is a term to 

which some other term has the relation R. Note that, by #14-21, both g ! R'y 

and ~g ! R'y imply E ! R'y. The contradictory of g ! R'y is not ~a ! R'y, 

^ — > — > 

but ~{[R'y] ■ a I -R'y}. This last would not imply E ! R'y, but for the fact 

— > 
that E ! R'y is always true. 

#3213. \-.R'y = x(xRy) [*32'11 . #2059] 

#32131. r . R'x = § (xRy) [#32-111 . #2059] 

#32132. h : CLRy . = . a = R'y . s . a = £ (*%) [#321-13 . #20*57] 
#32-133. r : /3Rx . = . £ - R'x . = . /3 = £ (*%) [#32-101-131 . #20-57] 

The use of #20*57 will in general be tacit. It happens constantly that we 
have propositions such as #32*13, in which a descriptive expression is shown 
to be identical with a class. In such cases, whenever the properties of the 
class are asserted of the descriptive expression, #20*57 is relevant. 

#32*14. *-:£ = £*.=-.£ = £ 
Bern. 

K #21*43. Dh::R = S.= 

[#32*1] s 

[#11*2] = 

[#20*25] = 

[#2015] = 

[#11*2] = 

[#21*43] = 



a Ry.= a , y .aSy:. 

a = x (xRy) . =., „ . a = & (xSy) :. 

(y):.a = x(xRy).= aL .a = ^(xSy)i. 

(y):fc(xRy) = x'(xSy):. 

(y) :. (sc) : xRy . = . xSy :. 

(x, y) : xRy. = . xSy :. 

J2 = £:rOKProp 

[Similar proof] 



#32*15. !-:£ = £. = .E = £ 

#32*16. h : R = S. = • $"-#"• = -R = 8 [#32*14*15] 

#3218. \-:xeR'y. = .xRy [#32*13 . #2033] 

#32*181. r- : y eB'« . = .xRy [#32*131 . #20*33] 

#32*182. h : x e~R'y . = .yeR'x [#32*18*181] 



SECTION D] BBFEBENTS AND BELATA OP A GIVEN TEBM 245 

The transformation from " xRy " to " x e R'y " is one commonly effected in 
language. E.g. suppose " xRy " is " x loves y," then " x e R'y " is " x is a lover 
ofy." 

#3219. \-'.RGS.O.R'yClS'y.%xCS<x 
Bern. 

K#32*18. Dt-:.Kv.3:xeR<y.D x .xeS'y: 

[#221] DzR'yCS'y (1) 

b . #32181 . D b :. Hp . D : y e R'x . D y . y eS'x : 

[*22\L] Drl^CS'* (2) 

I- . (1) . (2) . D I- . Prop 

#32 2. H : 4 sg R . = . A = E [#21-3 . (#32-03)] 

*32'201. b:AgsR. = .A=R [*213 . (*32'04)] 

*32'21. b.R = ag'R [#32-2. #303] 

#32-211. b .5= gs'iS [*32201 . #30-3] 

#3222. h.E!sg'i2 [*32'21 . *1421] 

*32221. h . E ! gs'JR [#32-211 . #14-21] 

#3223. b.8g'R=~R [#32-21 . *21-257] 

#32231. b . gs'R = R [#32-211 . #21-2-57] 

#3224. h.sg'5 = gs'i2 
Bern. 

b . #3223 . (#3201) . D b . sg'R = ty{a = % (xRy)} . 

[#21-33] Dha (sg<i?) y. = .a = x (xRy) . 
[*3111.*20'15] = . a = x (yRx) . 

[#32-101] = .aSi. 

[#32-211] = .a(gs'R)x (1) 

b . (1) . #11-11 . #21-43 . D b . Prop 

#32-241. h.gs'ii = sg<E [Similar proof] 

#3225. b : A sg R . = . A = 8g f R [#304 . #3222] 
#32-251. b:AgsR.~.A=gs t R [#304 . #32221] 
#32-3. b . {sg'(i2 A S)} ( y =~R'y n~S'y 
Note that we do not have 

8g'(R*S)=sg<Rnsg'S. 



246 MATHEMATICAL LOGIC [PART I 

Dem. 

h . *322313 . D h . {sg'(R n S)}'y = x{x(RnS)y] 
[*2333] = x~(xRy.xSy) 

[*22-39] = £> (arjRy) n £ (xSy) 

[*32-13] = £<y n"s<y . D I- . Prop 

*32 31. h . {gs'(12 n £)}<# = *R'x n S*ar 
*3232. l-.{sg'(i2wi8f)}'y=sS'ywii^ 
*3233. \-.{gs'(RvS)}'x = R'xvS'x 
*3234. r.{sg'(^#)}<2/ = -i?y 
*32'35. h . {gs'( - R)}'x = - S*« 

The proofs of the above propositions are similar to that of *32"3. 
*324. \-'..ElR'2. = :< 3L lR'z:x,yeR'z.'> Xiy .x = y [*30-21 .*3218] 

*32 41. h:.ElS'y.D:^'y = ^y. = .R ( y = S'y 
Dem. 

h.*4-86. Dh::xSy.= x .x = b:D:. 

xRy . = x . xSy : = v xRy . = x »x—b (1) 
h . (1) . #5*32 . D h :. xSy .= x .x = b: xRy . = x . xSy : = : 

xSy .= x .x—b:a>Ry .^ x .x = b (2) 
V . (2) . *10'11-281 . *3218-181 . D 

— > — ► 
H :.(a&) :xSy. = x .x = b :R ( y= S'yz = : (g;&) : #$y . ~ x . x = b : xRy . = x . x = b : 

[#30'3.*14-13] = : (36) : a% . e= x . a; = 6 : R'y = 6 : 

[*14-101] = :R'y = S'y (3) 

!- . (3) . *30-2 . D V :. E ! S'y . R'y =~8'y . = . R'y = S'y :. D > . Prop 
*3242. I- : . R'y = £<y . D : E ! R'y . = . E ! /S'y [*30-34 . *3218] 



*33. DOMAINS, CONVERSE DOMAINS, AND FIELDS 
OF RELATIONS 

Summary o/*33. 

If R is any relation, the domain of R, which we denote by T>'B, is the 
class of terms which have the relation R to something or other; the converse 
domain, d'R, is the class of terms to which something or other has the 
relation R; and the field, C'B, is the sum of the domain and the converse 
domain. (Note that the field is only significant when B is a homogeneous 
relation.) 

The above notations D'.R, d'B, C'B are derivative from the notations 
D, a, C for the relations, to a relation, of its domain, converse domain, and 
field respectively. We are to have 

C'B = £ {(ay) • ®Ry • v »-yRx) ; 
hence we define D, Q, C as follows: 

*33"01. T> = aR[u = ti{(ny).a;Ry\] Df 

*3302. a = £R[£=£{( a #).#%l] Df 

*33 03. C = yR [ 7 = & {(ay) : xRy . v . yRx}] Df 

The letter C is chosen as the initial of the word " campus." We require 
one other definition, namely of the relation of x to R when a? is a member of 
the field of R. This relation, which we will call F, is defined as follows: 

*33 04. F= $R {(ay) : xRy . v . yRx] Df 

We shall find that C= F. D will be the relation of a relation to its domain* 

D'a will be the class of relations having a for their domain. Similar remarks 
apply to Q. and C. The field of a relation is specially important in connection 
with series. 

The propositions of this number are constantly used throughout the 
remainder of the work. The ideas of the domain, converse domain, and field 
are very general, and have somewhat different uses for relations of different 
kinds. Consider first the sort of relation that gives rise to a descriptive 
function R'y. For this we require that R'y should exist whenever there is 
anything having the relation R to y, i.e. that there should never be more 
than one term having the relation R to a given term y. In this case, the 
values of y for which R'y exists will constitute the " converse domain " of R, 
i.e. <l f R, and the values which R'y assumes for various values of y will 



248 MATHEMATICAL LOGIC [PART I 

constitute the "domain" of R, i.e. D'R. Thus the converse domain is the 
class of possible arguments for the descriptive function R'y, and the domain 
is the class of all values of the function. Thus, for example, if R is the relation 
of the square of an integer y to y, then R'y = the square of y, provided y is an 
integer. In this case, d'R is the class of integers, and D'R is the class of 
perfect squares. Or again, suppose R is the relation of wife to husband; then 
R'y = the wife of y, d'R = married men, T)'R = married women. In such 
cases, the field usually has little importance; and if the values of the function 
R'y are not of the same type as its arguments, i.e. if the relation R is not 
homogeneous, the field is meaningless. Thus, for example, if »R is a homo- 

geneous relation, R and R are not homogeneous, and therefore "C'R" and "C'R" 
are meaningless. 

Let us next suppose that R is the sort of relation that generates a series, 
say the relation of less to greater among integers. Then D'R = all integers 
that are less than some other integer = all integers, d'R = all integers that 
are greater than some other integer = all integers except 0. In this case, 
C'R = all integers that are either greater or less than some other integer 
= all integers. Generally, if R generates a series, D'R = all members of the 
series except the last (if any), d'R = all members of the series except the first 
(if any), and C'R — all members of the series. In this case, "xFR " expresses 
the fact that # is a member of the series. Thus when R generates a series, 
C'R becomes important, and the relation F is likely to be useful. 

We shall have occasion to deal with many relations having some of the 
properties of series, and with many propositions which, though only important 
in connection with serial relations, hold much more generally. In such cases, 
the field of a relation is likely to be important. Thus in the section on 
Induction (Part II, Section E), where we are preparing the way for the con- 
struction of serial relations by means of a certain kind Of non-serial relation, 
and throughout relation-arithmetic (Part IV), the fields of relations will occur 
constantly. But in the earlier parts of the work, it is chiefly domains and 
converse domains that occur. 

Among the more important^ properties of domains, converse domains and 
fields, which are proved in the present number, are the following. 

We have always E ! D'R, E ! d'R, E ! C'R (*3312121122). (The last of 
these, however, is only significant when R is homogeneous.) 

*33 13. h : x e D'R . = . (gy) . xRy 

*33 131. I- : y e d'R . = . (ga?) . xRy 

*33 132. h:.xeC'R. = : ( 32 /) : xRy . v . yRx 

*3314. hixRy.D.xeD'R.yed'R 

*3316. h . C'R = D'R v d'R 



SECTION D] DOMAINS AND FIELDS OP RELATIONS 249 

$33*2*21 '22. The converse domain of a relation is the domain of its converse, 
the domain of a relation is the converse domain of its converse, and the field 
of a relation is the field of its converse. 

#33*24 \-:RlI)'Il. = .nl<I'K. = .'g L lC'R. = .'£lR 

♦33-4. b.T>'R = x{ftlR'x} 

with corresponding propositions (#33-41-42) for d'R and C'R. 
*33'43. b:El R'y.D.yea'R.R'yeV'R 
#33-431. h:(y).E!.R<y.D.09).£C(I'22 

#335. Y..C=~F 

#33'51. h : x e C'R . = , xFR 

The proofs of propositions concerning Q and G are usually similar to those 
for D, and are therefore often omitted. 



#3301. D~&R[a = x{(>xy).xRy}] Df 

#3302. a = /3jR[p=§{(<3a;).xRy}] Df 

#3303. C = yR[y = x {(ay) : xRy . v . yRx}] Df 
«33*04 F = xR {(ay) : xRy . v . yRx) Df 

•83-1. V : ciDR . = . a = x {(ay) • xRy] [#213 . (#3301)] 
#33101. \-:/3<IR.=:.l3 = g{(ftx).xRy} 
#33-102. h : yCR . = . 7 = x {(ay) : #-% • v . yRx] 
#33103. YuxFR.^-.i^-.xRy.y.yRx 
#33-11. b,. D'R = x {(ay) . *-%} [#33-1 . #30-3 . #20-59] 
#33-111. b.a'R = §{fax).xRy} 
#33-112. h . CJR = x {(ay) : ff-Ry . v . yRx] 
#3312. H.EID'jR [#3311 . #14-21] 

#33121. V.Eld'R 
#33122. H.E!(7'ie 

#33123. \-:aDR. = .a = T>'R [#304 . #3312] 

#33124 h:/3(LR. = .£==(KR [#304 . #33121] 

#33125. biyCR. = .y = C'R [*30\4 . #32123] 

#33-13. b:x€ D'R . = . (ay) • xRy [#3311 . *203-57] 
#33131. V :y ed'R . = .fax) .xRy 
#33132. V :. x e C'R . = : (ay) : xRy . v . yRx 
#3314 b-.xRy.D.xeD'R.yed'R 
Dem. 

h . #10-24 . D h :, Hp . D : (ay) ■ xRy : fax) ■ xRy : 
[#3313-131] D : x e D'i* . y e CPE :. D h . Prop 



250 MATHEMATICAL LOGIC [PART I 

*3315. b. ~R'y CD' R 
Dem. 

b . *3218 . >H -.xeR'y r D,. xRy . 
[*10-24] D x .fay).xRy. 

[*3313] D*. * e D'iS: 3 I- . Prop 

*33151. b.R'xCd'R 

*33152. b.~R'xv%xCC'R 
*33 16. I- . C<22 = D'R u O'E 

K*33'132.*10'42.D 

b ux e C'R . = : (gy) . xRy . v . (gy) . yRx : 

[*33'13~I31] = : x e D'R . v *a? ea'iJ : 

[*2234] =:x € I)<Rva'R (1) 

I- . (1) . *1011 . *20'43 . 3 f- . Prop 

*33f61. b.D'RCG'R.a'RCCR [*3316 . *2258] 

*3317. b:xRy.D.x,yeC'R [*33I4rl61] 

*33 18. f- : IXR = d'R . D . D'i2 = C'R 

Dem. 

b . *22*56 .5 b : D'R = d<fi . D . D<£ = D'R vd'R 

[*33 16] = C'R : Z> r . Prop 

*33*181. I- : d f # C D'R . = . D'.R = (?i2 

Dem. 

I- . *22-62 . 31- :d'R C D'R . = . D*R = D'R u <Pi2 

[*33*16] =C"i2:DKProp 

*33182. r:D*I£Ca<i2. = .(I<B = C?< J R [fimilar proof ] 

If R is the sort of relation which generates a series, so that "xRy" may 
be read "x precedes y" then G'R G D'R is the condition that the series may 
have no last term; since it states that every term which follows some term 
precedes some other term, and is therefore not the last of the series. 

*33-2. b.d'R = D'R 
Bern. 

b . #31-11 . #1011 . D b :xRy . = x . yRx : 

£*10-281] D I- : fax) . xRy . = . fax) . yRx : 

[*3313131] >H : y e d'R . = .ye D'R (1 ) 

I- . (1) .*1011 . #2043 . 5 b . Prop 

*B3VL b„D'R=<l?R [Similar proof] 



SECTION D] DOMAINS AND FIELDS OF RELATIONS 251 

*33'22. h.G'R^&R 

J}em. 

h . *33'16-2*21 .Oh. G t R=*Q*R\*'D , R 
[*33'16] = C'R . > I- . Prop 

*3324. ¥:^lT> l R. = .^ia t R. = .^lC t R. = .^lR 
Dem. 

V . *3313 . D H :. a rD<B . = r(a«) ::(&y) ~*% : 
[*25-5.(*ll-G3)} = :#!£ (I) 

K . *33131 . D F u 3 ! <FI2 . = : (fty> : <g*?) . *% r 
X*lI.-2] =:(aar,y)v«ftsf,: 

[*25'5] .ssfcT-R (2) 

H . *33-132. D h s: a lC'R.= :w(gr)s;<Ey)r-*jRy- v . yi^ :. 

h.(l).(2),(3),:>4-.Prop 

*33 241. h: D'lf = A . = ~OiR= A .3.^1.= A.,= . 12= A 
[*3324 . Transp . *2^51 . *25ol3 

*3325. KB^fn^CFEnM 

r- . *3313 . Oh :.. ic e ©*(# n S) . = : (gy> . at (R A-.S) y : 

[*21-33.*10-281] =z{*ay).xRy.xSyi 

[*I05] 3:toy.^^:(gy).a%: 

[*33'13] .- a e D'i? . a eD'fi : 

[*21-33] D^elKKnB'S (1) 

P.(l).*10 s ll. >f.-Prop 

*33'25i; h.af(jrn£)C..(P:Bn(FS [Similar proof] 

*33 252. h . C'(i2 n 8) C CjR rv £<£ [Similar proof] 

*33*26. h . D'(i2 vtS) = D'jB v-M 

h . *331&0 K :-. a? *©<(2£a £) .-= i<g^ v«{# «r3) # : 
[*23-3*.*10-281] ^'im} :*%..-»•. *% : 

[*H**2] = : <ay) ■ «% 1 v : ( a y) . ofify : 

[*3313] = t£e.]}fA.*i..*«£> f £? 

[*22-S4J =::-areEKK'uJ}?S ^> 

I- . (1} . *10I1 .^2043 JSr ..Prop 

*33<2G1. h . 0<(R o jSf)= Gf J2:"w G<$ [Similar proof] 

#33-262. h.0 f (i2crvS)W^^ w ^^ [*3S*2S-a61 i I'6} 



252 MATHEMATICAL LOGIC [PART I 

•33263. b-.RGS.l.D'RCD'S 

Dem. 

h . *23*1 . D h :. Hp . D : xRy . D x%y . xSy : 

[•10-28-27] D:(«):(aaf).«%.3-(ay).«flfy: 

[•3313] D:(x):xeD i R.D.X€D t 8: 

[•231] D:D'i2CD^:.3h.Prop 

•33-264. h-.RGS.l.a'RCa'S [Similar proof] 

•33-265. t-'.RGS.O.C'RCC'S [*33-263-26416 . •2272] 

•33-27. \-.C'R = D'(RvR) 
Dem. 

r . *3316-2 . D h . C'R = D'l* w D'12 

[•33-26] « D<(22 c»i).DI-. Prop 

•33-271. h . C'R = (!<(£ a E) [Similar proof] 

•33-272. H.D'(/2u5) = a'(EciS) = 0'(Ec»S) = 0'i2 [*33*27-271'16] 
•3328. KD'V = <I<V = (7'V = V 

Dew. 

h . *10'25 . •25-104 .Dh:.(«): (gy) . xYy :. (a;) : (gy) . yVa? :. 

[•33-13131] D h :. (*) . x e D'V : (#) . x e d't :. 

[•24-14] Dh:D<V = V.<I<V = V (1) 

[•33-16] 3l-.(7<t =VuV 

[•22-56] =V (2) 

h.(l).(2).Dh.Prop 

•33-29. r.D'A = (I'A = C"A = A [*33'241 . #21-2] 

•333. b:.aCT><R. = :xea.O x .>&lR<x 
Bern. 
h . *32-181 . D H :. x e* . D x . a ! i2'# : = : x ea . D. . (#y) . #ify : 
[•3313] = :xea.D x .xeD'R:.Db.¥rop 

•33-31. r:./3C(KR. = :ye/3.:> 2 ,.g! J K'y [Proof as in *33 3] 

The three following propositions are used in the theory of selections (*80, 
•83 and *85). The second of them is also used in the theory of greater and 
less (*117) and in the theory of transitive relations (*201). 

•3332. \-:D t RnT>'S = A.0.R*S = A 

The converse of this proposition is not true. 

Dem. 

h.*23'33. D\-:x(RnS)y.D.xRy.xSy. 

[*33-14.*22-33] D.xe D'R r> V<S . 



SECTION D] DOMAINS AND FIELDS OF RELATIONS 253 

[#1024] O.RlD'RrsD'S (1) 

K(l).Transp. D H:D'i2nD'^ = A.D .~~{x(RKS)y} (2) 

\-.(2).*ll-ll'3.3b:WRrxI)'S=A.'}.(x,y).~{x(RnS)y}. 
, [#25-15] D.22n£=A:3KProp 

#33 33. I- : Q<R n a'-Sf=A.D.iJ^=A [Proof as in #33*32] 

#3334. h:C'i2nC'£=A.:>..RA£=A 
Dem. 

r . #33161 . #2249 . D K D'E n D'S C C'R nC'S. 

[#2413] Df-:(7'i2n(7^ = A.D.D'i2nD'5f = A. 

[#3332] D . i* n £= A : D h . Prop 

#3335. \-:.& t RCa. = :xRy.'2 X) y.xea 

Dem. 

h.*33'13.Db:.D t RCa. = :(^y).xRy.D x .xea; 

[#1023] = : xRy . D., y . x e a : . D h . Prop 

#33-351. >:. CI'E C a. = :xRy.D x , y .yea [Proof as in #3335] 

#33352. Yi.C'RCa.^ixRy. D x>y .x,yea 

Dem. 

K #33-16. #22-59. D 

r:.(7'.RCa.= :D<.RCa.(I'i2Ca: 

[#33'35'351] = : xRy .D, ir *ea: #.% .0 XiV .yea: 

[#11-391] = :xRy.D x>y .x, i/ea:.D h . Prop 

The two following propositions (#33-4*41) are very frequently used. 

#33-4. h.T>'R = x\Rl%x} 

Dem. 

h . #33-13 . D h : « eD'lS . = . (ay) . «% . 

[#32181] =.(>&y).yeR'x. 

[#24-5] ■ ' s.alS"'* " <1) 

I- . (1) . #10-11 . #20-33 . D h . Prop 

#33-41. \-.(I'R = §falR'y} [Similar proof] 

#33-42. \-.C'R=x{'&l(R'xvR'x)} 
Dem. 

\- . *33'4-41-16 > D I- . C'R = & {a ! i2^} w % {a I .#•#} 

[#22-391] = & [a J -R'* ■ v . a * R ' x ) 

[#24-56.*20'15] = & {3 ! (2S<# u £<#)}. DK Prop 



254 MATHEMAKOAL LOGIC [PART I 

*3343. t-iElR'y.D.yna'R.M'yeD'R 
Bern. 

. r . *S0-32 ,0 r : E I R'y . D . (R'y) Ry . 

[*3314] O.yea'R.R'yi-D'RzDb.Prov 

*33431. H : (y) . E tjBty.O . (#) . C d<R 

Dem. 

r . *3343 . D h :. Hp .O zyed'R . 

[Simp] Oiyep.O.yed'R (1) 

h . (1) . *10-1 1-21 .O f : Hp . 3 . £ C d'R (2) 

K(2).*10ir21.3f.Prop 

*33-432.h:(^).E!i^.3.<Pi2 = V 

Pew. 

f- . *3343 . *1011-27 . 3 h : Hp .3 . (y) .ye d'R- 

[*2414] 3 . <PI2 = -Y .: O h . Prop 

*3344. H^l^.D.^eD'E.^ca^ 
Dew. 



\j \j 



f . *3343 ^ . 3 4- r: .Hp . 3 . # e (I'^R . 22<a> e T)<R . 

[#33'2-21] 3 . x e D'R . R'x e d'R : 3 K Prop 

*33 45. buy e d*R vd'S .D y . R'y = S'y:5 ,R = S 

Note that by our conventions as to denoting expressions, the scope of 
both R'y and S'y in the above is " R f y = S'y" and I2'y is to be first 
eliminated. 

Z)em. 
I- . *3011 .Of ::R'y = S'y . = :- (g6) : xRy.= x .x = b:b'=8 i y :. 
[#30*11] = :. (g6):.« J By.= a; .a/=5;.(?3c):^|/«= a .,. x = c : 6 = c:. 

[#131 95] ■ = : . {fib) : xRy, = x .x — b: xSy . =3.". x = b : . 

[*10322] 3 :. xRy . ~ x . <cSy (1) 

I- . (1) . 3 I- :: Hp . 3 :. y e d'R vd'S.D: xRy . = . xSy :. 
[*532] 3 :. y e <1'R v d'S . xRy.=. yed'Rvm® . xSy*.. 

[*3314.*4'7l] 5:.xRy. = .xSy (2) 

H . (2) . *llll-3 . 3 J- :. Hp . 3 : (x, y) : xRy .= .xSy. 
[*2143] 3:.K = £:.3f-.Prop 

*33 46. f :. x e D'R u D'S .3*. R'x = S'x:D.R = S [Proof as in *33i 

*3347. f i.yed'R u d'S .D y .R'y = ~S'y : 3 . R = S 
Dem. 

b . *33"41 . Transp .Db:y~e d'R v d'S . 3 . Tl'y = A . &y = A 



SECTION D] DOMAINS AND FIELDS OF RELATIONS 255 

f-.(l).*13172.*4'83.Df-:Hp.D.(y).^="^y. 

[#30-41] D.R=!s. 

[#3214] D.J? = £OKProp 

#3348. Vi.xe D'R u T>'S . D* . R'x = S'x : D . R = £ [Proof as in #33-47] 

#33 5. r.C=2? 
Bern. 

h . #321 . •> h :. o?i? . = ,a = $(xFR) 
[#33103] = % {(&y) : xRy . v . yRx) . 

[#33-102] = . aCR (1) 

h . (1) . ■#11-11 ... #2143 . D H . Prop 

*3351. b ix €&£.. = . xFR [*33132103} 

F is useful in ordinal arithmetic, where we are concerned with a series 
generated by a relation P, and " xFP " expresses the fact that a; is a member 
of this series. The above two propositions (*33'5 - 51) will be much used in 
Part IV, where we deal with the foundations of ordinal arithmetic, but will 
not often be referred to elsewhere. 

#33-6. r : R e%a. = .a = B'R 
Bern. 

V , #32-181 .0 I- : ReD'a. = . aDR . 

[#33123] = .« = D'jR:>KProp 

#33-61. f- : R e CT <a . = . a = d'R 
*33 62. r : R e ^a . = . a = O'tf 



*34. THE RELATIVE PRODUCT OF TWO RELATIONS 

Summary q/**34. 

The relative product of two relations R and 8 is the relation which holds 
between x and z when there is an intermediate term y such that x has the 
relation R to y and y has the relation 8 to z. Thus e.g. the relative product 
of brother and father is paternal uncle; the relative product of father and 
father is paternal grandfather ; and so on. The relative product of jR and # 
is denoted by " i2 1 S " ; the definition is : 

*3401. R\S = tcz{(>g L y).xRy.ySz} Df 

This definition is only significant when Q'R and D'# belong to the same 
type. 

The relative product of R and R is called the square of R ; we put 
*3402. £ 2 = £|.R Df 
*3403. R S = R 2 \R Df 

The most useful propositions in the present number are the following : 

*34-2. t-.Cm'(R\S) = S\R 

I.e. the converse of a relative product is obtained by turning each factor 
into its converse and reversing the order of the factors. 

*3421. h.(P\Q)\R = P\(Q\R) 

I.e. the relative product obeys the associative law. 

*3425. b.P\(QvR) = (P\Q)v(P\R) 

*34'26. r.(PvQ)\R = (P\R)v(Q\R) 

I.e. the relative product obeys the distributive law with respect to the 
logical addition of relations. (For logical multiplication instead of logical 
addition, we only get inclusion instead of identity; cf. *34 # 23'24.) 

*3434. \-:RGP.SGQ.3.R\S<ZP\Q 

*34 36. h . D'(P | Q) C D'P . d'(P j Q) C ti'Q 

*34'41. r : E ! P'Q'z . D . P'Q'z = (P | Q)'z 



*3401. R\8 = xz{{^y).xRy.y8z} Df 
*3402. R* = R\R Df 

*3403., R S = R 2 \R Df 

#341. \-:x(R\S)z. = .(Ry).xRy.ySz [*21-3 . (#3401)] 



SECTION D] 



THE RELATIVE PRODUCT OF TWO RELATIONS 



257 



#3411. h:x(R\S)z.= .Rl(R<xnS<z) 
Bern. 

h . #341 . #32-18181 . D 

t-:x(R\S)z. = .(Ky).y € R<x.yeS<z. 
[#2233] = .(Ry).ye*R'x*~S'z. 

[*24-5] = . a ! (R'x * ~S'z) : D h . Prop 

*3412. \-.R\S = a&fal(R<xrs~S<x)} [#21-33 . *3411] 

#34-2. \-.Cnv'(R[S) = S\R 
Bern. 

r . #31131 . D h : x [Gnv'{R \ S)} z . = . z (R \ S) x . 

[*34i] =-(wy)- z Ry-ySx- 

[«31'll] =-(ay)-y^.a?Sy. 

:■ I* 34 * 1 ] s.ar(S|^)« (1) 

r . (1) . #1111 . #21-43 . D r . Prop 

#34-202. \-.R\S = (Cnv'R) \S 
Bern. 

\-.*3ll31.Dh:x(Cnv'R)y.ySz. = .yRx.ySz. 
[*31H] =.xRy.ySz (1) 

H . (1) . #10-11-281 . #34-1 . D h : * {(Cnv'i) \S} z . = .x(R\8)z (2) 
f- . (2) . #1111 . #21-43 . D h . Prop 

#34-203. h.R\S = R\ (Cnv'S) [Similar proof] 
#34-21. \-.(P\Q)\R = P\(Q\R) 



Be 



m. 



■ (a*) : (ay) • a^y ■ yQz ■• zRw 

■ (ay)- ®Py ' (a*) • yQz.zRw 
-('3.y).xPy.y(Q\R)w (1) 



H^S^l^lO^l.DI-r^a^.^PIQ^.^^.^ 
[#11-6] = 

[*34-l.*10-281] = 

h . (1) . #11-1 ■ . #341 . #21-43 . D h . Prop 

#34-22. P1Q\R = (P\Q)\R Df 

This definition serves merely for the avoidance of brackets 

#34-23. \-.P\(Q*R)G(P\Q)n(P\R) 

Bern. 

h . #341 . D 



V:.x{P\(Q«R)}y.= 
[#23-33] = 

[#10-5] D 



R&W I 



(Rz).xPz.z(Q*R)y: 

(3^) • asPz . sQy . zRy : 

(3^) • #P* . zQy : (jj*) . xPz . zRy: 



17 



258 MATHEMATICAL LOGIC [PART I 

[#341] 3:x(P\Q)y.x(P\R)y: 

[*23-33] D:x{(P\Q)r*(P\R)}y (1) 

h. (1). #1111. 3 h. Prop 
The converse of the above is not true. 

#3424. b.(PnQ)\RG(P\R)n(Q\R) [Similar proof] 

#3425. H .P | (Qw _R) = (P | Q)w(P | i2) 
Dem. 

h . #2334 . *10-281 . D 

f- :. (as) . aPs . « (Q v R) y . = : (gs) : aPs : zQy . v . zRy : 
[*4-4.*10*281] = : (gs) : xPz .zQy.v. xPz . zRy : 

[#10-42] = : (g^) . xPz . zQy : v : (g*) . xPz . zRy : 

[#341] = : x(P j Q)y . v . x{P \ R)y : 

[*23-34] =:x(P\QvP\R)y (1) 

H '. (1) . #1111 . #341 . D h . Prop 

#3426. h . (P a Q) | R = (P | R) v (Q \ R) [Similar proof] 

The above two forms of the distributive law, and the associative law 

(#34*21), are the only ones of the usual formal laws that hold for the relative 

product. The commutative law, in particular, does not hold in general. 

#3427. h:i2 = P'.D.P|P = P'|P 
Dem. 

h . #21*43 . D h :. Hp . D : (x, y) : xRy . = . xR'y : 
[*1 1*401] D : (x, y) : xRy . yPz . = z . xR'y . yPz : 

[#10*281] D : (x) : (gy) . xRy . yPz . = z . (gy) . xR'y . yPz : 

[#21*15] D: J R|P = E'|P:.DI-.Prop 

#3428. h:R = R'.3.P\R = P\R' [Similar proof] 

#34*29. h:R = R'.D.P\R\Q = P\R , \Q 

Dem. 

h .#34*27 . D h : Hp . D . R \ Q = R' \ Q. 

[#34*28] D.P\R\Q = P\R'\Q:1\-. Prop 

In proving the equality of two relations, say R and S, we usually establish 
first an asserted proposition of the form 

xRy . = . xSy 
or Hp . D : xRy . = . xSy. 

We then proceed by #11 11 (together with #11*3 in the second case) to 

(x, y) : xRy . = . xSy or Hp . D : (x, y) : xRy . = . xSy, 
whence the result follows by #21 43. We shall in future omit these steps, 
and write " D h . Prop " after we have established 

xRy . = . xSy or Hp . D : xRy . = . xSy. 
A similar ellipsis will be made in proving the equality of classes. 



SECTION D] THE RELATIVE PRODUCT OF TWO RELATIONS 

#34 3. h : a ! (P I Q) . = . a ! (Q'P n D'Q) 

Dem. 

h.*25-5.D 
F::a!(P|Q).= 
[#341] = 



259 



[#11-27] 

[#11-24] 

[#11-27] 

[#11-54] 

[*3313131] 

[#22-33] 

[#24-5] 



■•(a«»y)-*(-P|Q)y!- 

-('&x>y)'('&z)>xPz.zQy;. 

.(Rx,y,z).xPz.zQy:. 

.(Qz,x,y).xPz.zQy:. 

■ (a*) : - (a#> y) • xPz • z Qy •• 

■ (3*) : - (a«) • <*>Pz : (gy) . sQy :. 

:(Kz):.ze<I<P.zeI>'Q:. 

.(Sz):.zea<PnT>'Q:. 

.3 !((TPn D'Q) :::>!-. Prop 



#34 301. h : CFP n D'Q = A . = . P | Q = A [#343 . Transp] 

#34302. h : C'P n C'Q = A. D .P | Q = A . Q | P = A 
Dem. 

!-.*3316.DI-:Hp.D.<I'PnD'Q = A.a'QnD'P = A 
[#34-301] D . P | Q = A . Q I P = A : D K Prop 

^ar(P|Q).:>.3!P.a!Q 



#3431. 

Dem 



#34-32. 

#3433. 

Dem. 



#34-34. 

Dem. 



h . #34-3 . D h : Hp . D . a ! (d'P r> D'Q) . 
[#24-561] D.Ria'P.RlD'Q. 

[#33-24] D.a!P.a!Q:DI-. Prop 

l-:.P = A.V.Q = A:D.P| Q = A [#3431 . Transp . #2551] 
V:xe\) i R.~.x{R\R)x 

h . #3313 . D h : a; e D' # . = . (ay) . #% . 
[#4*24] = . (ay) . xRy . xRy . 

[#31-11] =.(^y).xRy.yRx. 

[#34-1] = .tfCK|.R)*;:DI-.Prop 

h:i2GP.£GQ.D.i2|£GP|Q 

H.*23-l.Dh:.Hp.D:a? J Ry i D B ,„.a;Py:y&.D ytI .yQ«: 

[*11-2.#10-1-41] D : «% . D . #Pt, -.ySz.D. yQz : 

[#3-47] D-.xRy.ySz.D.xPy.yQz (1) 

h.(l).*10-ll-21-28.D 

l-:.Hp.D:(ay).^%.2/^.D.(ay).^Py.^: 
[#34-1] D:*(i2|fl)^.D.a?(P|Q)« (2) 

h.(2).*llll-3.Dh.Prop 

17—2 



260 MATHEMATICAL LOGIC [PART I 

#34 35. h : a ! R . d'R C D'P . D . a ! R \ P 

Bern. 

h. #33-24. DHHp.D.ald'iZ (1) 

1- . #22621 . D h : Hp . D . d'R = d'R n D'P (2) 

b . (1) . (2) . D h : Hp . D . a ! d'R n D'P . 
[#34-3] D . a ! E j P : D I- . Prop 

*34351. h-.alP.D^Ca^P.D.aSPjP [Proof as in *34'35] 

#34 36. h . D'(P | Q) C D'P . <J'(P j Q) C d'Q 
Bern. 

b . #3313 . D H :. «eD'(P | Q) . Z> : (a*) . #(P | Q)z : 
[#341] D:(a*,y).*Py.yQ*: 

[#11-23] D:( W ,z).xPy.yQz: 

[*ll-55.*10-5] D : (ay) . #Py : 

[#3313] Dr^eD'P (1) 

Similarly I- :.se(I'(P j Q) . D : zed'P (2) 

h . (1) . (2) . #1011 . D H . Prop 
The following proposition is a lemma for #95'31. 

#34*361. h : a ! R . D'P C d ( P . d'R C D'Q . D . a I P | P | Q 
Dew. 

h.*34-35.DI-:Hp.D.a!PiQ (1) 

K*34-36.Dh:Hp.D.D'(P|Q)C<I'P (2) 

K (1). (2). #34-351. DK Prop 

#34-37. h . C'(P | Q) C D'P v d'Q [#34-36 . #33161 . #2272] 

#34-38. h . C'(P | Q) C C"P u C Q [#3437 .#33161 . #2272] 

#34-4. b:b = P'c.c = Q'z.D.b = (P\ Q)'z 

Bern. 

b . #3031 . D I- : Hp . D . bPc . cQz . 

[#34-1] D.b(P\Q)z (1) 

K #30-31 . Dh:.Hp.D: yQz.D y .y = c: 
[Fact] D : #Py . y Qz . D«, y . #Py .y=c. 

[#1313] Da.y.tfPc (2) 

h . #30-31 . >h :. Hp. D : a;Pc . D x . cc= b (3) 

b.(2).(3).Db:.H.p.D:ccPy.yQz>D x>y .w = b: 
[#10-23] D : (ay) • xPy . y0* ■ =>. ■ » = 6 : 

[#341] D:a?(P|Q)^.D a; .«=6 (4) 

h . (1) . (4) . #30-31 . D b . Prop 
#34-41. r : E ! P'Q'z . D . P'Q's= <P | Q)'z 



Bern. 



b . #30-52 . D h : Hp . D . ( a 6, c) . 6 = P'c . c = Q'z . 
[*30-51.*34-4] D . (a&) • 6 = P'Q'* • b = (P | Q)'s . 

[#14145] D . P'Q'z ={P | Q)'* : ,3 h . Prop 



SECTION D] 



THE RELATIVE PRODUCT OF TWO RELATIONS 



261 



The above proposition is no longer true if we change the hypothesis into 
E ! (P | Q)*z, since (P | Q)'z may exist when P'Q c z does not. Suppose, e.g., 
that Q is the relation of child to father, and P the relation of daughter to 
father. Then (P | Q)'z = the granddaughter of z, but P'Q'z = the daughter of 
the child of z. The first exists whenever z has only one granddaughter, 
while the second requires further that z should have only one child. 

For the same reason we do not have 

b = (P\Qyz.D.(Rc).b = P'c.c = Q'z. 
This will hold if P,Q are one-many relations (cf. *71), but not in general 
otherwise. 



*34-42; 
Dem. 



*34-5. 
*34-51. 

Dem. 



\-:(z).R'z = P'Q'z.D.R = P\Q 

h . *14-21 . D h :. Hp . D : (z) . E ! R'z : (z) . E ! P'Q'z 
h . (1) .*3441 . D h :. Hp . D : (z) . R i z = (P \ Q)'z : 
[*30-42.(l)] D:P = P|Q:.Dh. Prop 

h : xR?y . = . (<&z) . xRz . zRy [*34-l . (*3402)] 

V : xR 3 y . = . (qz, w) . xRz . zRw . wRy 

h . *341 . (*34-03) . D 



(1) 



I- :.xR 3 y . = 
[*34-5] = 
[*ll-55] = 
[*ll-2] = 
b.R s = R\R* 

k : a ! r? . = . a ! d<p n <pp 



*3452. 
*3453. 
*34531 
*3464. h : xRx . D . #P 2 # 
Dem. 



(gw) . #jR 2 m/ . wRy : 

(gw) : (a^) . xRz . zRw : wRy : 

(g[w, #) . xRz . zRw . wRy : 

(a^, w) . #jR.z . zRw . wRy :. D H . Prop 

[*34'21] 

[*34-3] 

[*3453 . Transp] 



*3455. 
*3456. 
*346. 

Dem. 



h . *4'24 .Oh: xRx . D . xRx . xRx . 
[*10-24] D . (ay) • xRy . yRx . 

[*34-5] D . xR?x : Z> h . Prop 

h : . P 2 C S . = : #Py . yRz . D*, j,, , . xSz [*34'5 . *10-23] 

h . D'P 2 CD'iJ . (I'P 2 C d'R . G l R? C C'R [*343638] 

I- . (R rx Sf G P 2 n £ 2 



h . *34-5 . D h:.a!(iJn%.= 
[*23-33.*10-281] = 

[*4-3.*10'281] = 



(rz) .x(RnS)z.z(RnS)y: 
(a^) • xRz . xSz . zRy . zSy : 
(a^) • xRz . zRy . xSz . zSy : 



262 MATHEMATICAL LOGIC [PART I 

[*10"5] D : faz) . xRz . zRy : (ftz) . xSz . zSy : 

[*34-5] D : xB?y . xS*y : 

[*23-33] 1:x(R*-nS*)y (1) 

K (1). #11-11. DH. Prop 

#3462. \-.(RvSf = R*\jR\SvS\RvS* 

Dem. 

h . #34-26 .3b.(RvSY = R\(RvS)vS\(RuS) 

[#34-25] = R*vR\SvS\RvS 2 .D\-. Prop 

The above proposition is a lemma for #160 - 51, as is also #34*73, which 
employs the above proposition. 

#34-63. h . Cnv'(l^) = (Cnv'E) 2 

Dem. 

K #31-131. D 

I- :. x {Cnv'(R 2 )} y . = : yR*x : 

[#34'5] = : (gz) * y^ . sifo; : 

[#31-131.*10-281] = : (rz) . xRz . zRy: 

[#31-131. #34-5] =:^(Cnv f E) 2 2/:Dh.Prop 

#34-7. h.Cnv'OSf|£) = £jS 
Dem. 

H . #34-2 . D I- . Cnv<(£ | S) = (Cnv'S) 1 8 
[#34-202] = £|£.Dh.Prop 

Thus S | # is always a symmetrical relation, i.e. one which is equal to its 
converse. 

#34-701. h.Cnv<(£|£) = &|# [#34-2-203] 

#34-702. h.C"(£|£) = D<>Sf 
Dem. 

h . #34-37 . D h . C"(£ j 5) C D<£ u (I'S 

[#33-21] C D<£ (1) 

h . #33-13 . D h : * e D'S . D . (gy) . xSy . 

[#31-11] D.( a2 /).^.^. 

[#34-1] 3.a?(£|#)«. 

[#33-17] D.*e(7'(/8f|5) (2) 

l-.(l).(2).*10-ll .Dh.Prop 

#34-703. \-.C'(S\S) = a ( S [Similar proof] 



SECTION D] THE RELATIVE PRODUCT OF TWO RELATIONS 263 

*3473. H : C'P n O'Q = A . D . (P <y Qf = P 2 c; Q 2 
Dm. 

f-.*34'302.3h:Hp.D.P|Q = A.Q|P = A. 
[*25'24] D.P 2 oQ 2 = P 2 uP|QvyQ|PuQ 2 

[#34-62] = (P o Qf : D H . Prop 

#348. \-:R = R.R><ZR.O.R = R* = R\R 
Dem. 

H. #34-28. Dh: R = R. 3.R? = R 1 5 (1) 

I- . #3433 . #3314 . D h :a%. D.x(R\R)x (2) 

h.(l).(2). DH.iZ^.Dna-ity.D.tfifr** (3) 

h . (3) . #231 . D\-:.R = R.R*GR.D: xRy . 3 . #ifa : 

[#4*7] D : #ity . D . #.&« . #ify . 

[*10-24.*34-5] 2.xR 2 y (4) 

r.(4).*llll'3. DH:Hp.D.#G£ 2 (5) 

h.*327. Dh:Hp. D.i^Gfl (6) 

l-.(5).(6).*23-41.D!-:Hp.D. J B=i2 2 (7) 

l-.(l).(7). DKProp 

The hypothesis of the above proposition is the hypothesis that R is 

symmetrical (R = R) and transitive (R 2 G R). These are the formal properties 
of those relations which can suitably be regarded as expressing equality in 
some respect. 

#3481. \-:R = R.R 2 GR. = .R = R.R? = R [#34-8 . #4-71] 

The following propositions are lemmas for #34'85, which is used in #72*64, 

#3482. h :. R = R. R 2 GR.O: x € D'R. = .xRx 
Dem. 

K #3433. 3\-:x€D'R. = .x(R\R)x [1) 

K*34-8. D\-:.TIp.D:x(R\R)x. = .xRx (2) 

h.(l).(2).Dr.Prop 

*3483. \-:R = R.R?GR. xRy .D.R'x = R'y 

Bern. 

K #31-11. 0\-:.Kp.D:yRx: 

[#3'2] D : xRz . D . yRx . xRz . 

[*34-55.Hp] l.yRz (1) 

h.*3-2. ^[■•..H^.^-.yRz.'D.xRy.yRz. 

[*34-55.Hp] D.xRz (2) 

h . (1) . (2) . D h :. Hp . D : xRz . = .yRz : 

[*1011-21.*2015.*32111] D : R'x = R'y :. D h . Prop 



264 MATHEMATICAL LOGIC [PART I 

*3484. b:R = R.R*GR.y € B<R.R<x==R'y.'2.xRy 
Bern. 

K #34-82. Dh:Hp.D.y% (1) 

K #32181 . #20-31 . D h :. Hp . D : xRz . =, . yRz : 
[#10-1] . D : xRy . = . yRy (2) 

h.(l).(2). Dt-.Prop 

#34-841. h:R = R. R? G J? . xeD<R .%x = %y . D . xRy 
Bern. 

h . #34-84 ^^ . D h : Hp-. D . v^ . 

xv ' ' 

[*31-ll.Hp] D.xRy.Dh. Prop 

#34-85. h:.R = R.R*GR.3:xRy. = .xeD'R ."R'x^&y 
[#34-83-841 . #33-14] 



*35. RELATIONS WITH LIMITED DOMAINS AND 
CONVERSE DOMAINS 

Nummary o/#35. 

In this section, we have to consider the relation derived from a given 
relation R by limiting either its domain or its converse domain to members 
of some assigned class. A relation R with its domain limited to members of 
a is written "a^R"; with its converse domain limited to members of /3, it 
is written "R[fi"; with both limitations/it is written "a ^R^fi." Thus 
e.g. "brother" and "sister" express the same relation (that of a common 
parentage), with the domain limited in the first case to males, in the second 
to females. "The relation of white employers to coloured employees" is a 
relation limited both as to its domain and as to its converse domain. We put 
*3501. a *\R = % (x e a . xRy) Df 
with similar definitions for R\ a and a ^ R[ /3. 

A particularly important case is the case in which the same limitation is 
imposed on the domain and on the converse domain, i.e. where we have a 
relation of the form "a 'J 22 fa." In this case, the limitation to members of a 
may be more briefly stated as being imposed on the field. ITor this case, it is 
convenient to adopt "R£a" as an alternative notation. This case will be 
considered in #36. 

It is convenient to consider in the present connection the relation between 
x and y which is constituted by x being a member of a and y being a member 
of £. This relation will be denoted by "a f .£." Thus we put 
*3504. af p = x~§{xea.ye&) Df 

The chief importance of relations with limited fields arises in the theory of 
series. Given a series generated by a relation R, let a be a class consisting 
of part of this series. Then a is the field of the relation a "\ R fa or K£ a, and 
it is this relation which is the generating relation of the series of members of 
a in the same order which they have as parts of the original series. Thus parts 
of a series, considered not merely as classes but as series, are dealt with by 
means of serial relations with limited fields. 

Relations with limited domains are not nearly so much used as relations 
with limited converse domains. Relations with limited converse domains play 
a great part in arithmetic, especially in establishing the formal laws. What 
is wanted in such cases is a one-one relation correlating two classes or two 
series. That is, we want a relation such that not only does R'y exist whenever 

yi-U'R, but also R'x exists whenever x € D'jK. The kind of relation which is 
most frequently found to effect such a correlation is some such relation as D 



266 MATHEMATICAL LOGIC [PART I 

or G. or G, or some other constant relation for which we always have E ! R'y, 
with its converse domain so limited that, subject to the limitation, only one 
•value of y gives any given value of R l y. Thus for example let X be a class of 
relations no two of which have the same domain; then DfA, will give a one- 
one correlation of these relations with their domains: if R, Se\, we shall have 

T>'R = D'S.D.R = S. 
We shall also have D< R = (D[ \)'R and B'S = (D f \)'S. Moreover the con- 
verse domain of D [\\ is \, and the domain of D f X is the class of domains of 
members of \. Thus D[\ gives a one-one correlation of \ with the domains 
of members of X. It is chiefly in such ways that relations with limited converse 
domains are useful. 

For purposes of reference, a great many propositions are given in the 
present number, but the propositions that will be used frequently are com- 
paratively few. Among these are the following: 
*35-21. \-. a ^\R[0 = ( a J \R)\-0 = a J \(R[0) 
#35-31. h.(R[a)[0 = R[(an0) 
#35354. r.(£ra)|£=.R|a1# 

I.e. in a relative- product it makes no difference whether we limit, the 
converse domain of the first factor, or the domain of the second. 

#35412. \-.R\-(0v0') = R\-0vRt0' 

#35452. b-.a ( RC0.D.R[0 = R 

#35 48. b:a<PCa.D.P\(a1R) = P\R 

#35 52. r . Cnv'(R f 0) = 1 R 
#35 61. h . D'(a 1 R) = a n D'R 
#3564. r . a<(R [ 0) = n d'R 
#35 65. V : 0Q(1<R .1 .<1'(R\ 0) = 

The hypothesis C d'R is fulfilled in the great majority of cases in which 
we have occasion to use R[ 0. 
#35 66. \-:<I<RC0. = .R\-0 = R 
#357. \-:^{(R[0Yy}. = .ye0.cf>(R t y) 

This proposition is used very frequently, owing to the fact that limitation 
of the converse domain is chiefly applied to such relations as give rise to 
descriptive functions (e.g. D, Q, G). 
*35-71. r :.ye0.3 y .R'y = S'y :3 . R[ = S[ 

This proposition is useful for a reason similar to that which makes #35*7 
useful. 

#3582. \-.a'[0=a J \Y[0 

Owing to this proposition, the properties of a \ can be deduced from the 
already proved properties of a "\ R \ 0, by putting R = V. 



SECTION D] LIMITED DOMAINS AND CONVERSE DOMAINS 267 

The relation "a f @" is what may be called an "analysable" relation, i.e. it 
holds between x and y when xea and ye^, i.e. when x has a property inde- 
pendent of y, and y has a property independent of #. 

#35 85. I- : a ! £. D . D'(af £) = a 
#3586. h : g ! a . D . (I<(a T £) = £ 

If either a or fi is null, so is a f /3 (#35-88). 



#3501. a1-R = $0(a?ea.fl!JRy) Df 

#35-02. R\ft = mxRy.yeP) Df 

#35-03. a J \R\-/3 = x§(xea.xRy.y € 0) Df 
#35-04. a\P = %$(xea.ye&) Df 

#3505. £<# t /3 = (fl'a>) t £ Df 

The last definition serves merely for the avoidance of brackets. 
#35-1. V:x(a J \R)y. = .xea. xRy [#21 3 . (#35 01)] 
#35101. h:x(R\-j3)y. = .xRy.ye/3 
#35102. V-.xia^R^^y.^.xea.xRy . y e /3 
#35103. \-:x(a^ 0)y. = .xea.yej3 
#3511. Ka'|.R| k £ = (a1jR)«(.R| k /8) 
Dem. 

h . #35-102 . D h : x(a] R\P)y . = . xea .xRy. y e/3 . 

[#424] = .xea. xRy . xRy . y e y8 . 

[*35-l-101] -.^(al^y.afCief^y. 

[#23-33] = . x {(a 1 22) r> ( R f £)} y : D J- . Prop 

#3512. \-.(a J \R)n(S\'/3) = a'\(RnS)tl3 
Dem. 

>.#23-33.Dr-:a;{(a1i2)n(^f/5)}y.s.«(a1i2)y.a?(^f ; /3)y. 

[#35-1-101] = .xea.xRy.xSy.ye/3. 

[#23-33] = .xea.x(R*S)y.ye0. 

[#35-102] =.a?{a1(i2i s »i8f)| k )8}y:Dh.Prop 

#35-13. \-.(a J \R)n(/3 J \S) = (an/3) J \(R*S) 
Dem: 

\-.*23-33.O\-:x{(a J \R)*(0 J \S)}y. = .x(a J \R)y.x(/3 J \S)y. 

[#35'1] = .xea. xRy . x e /3 . &v% . 

[*22-33.*23-33] = .xe(a n /3) ..*(# A £)y . 

[#35-1] = . a {(a n £)1 (R n S)} y : D h . Prop 

#3514. h.(R\-a)n(St/3) = (RnS)t(a*l3) [Similar proof to #3513] 



268 MATHEMATICAL LOGIC [PART I 

*3515. ^.(«\Rt/3)n(c(1Stl3') = (anay'\(RAS)t(l3rsl3') 
Bern. 
K*3511.0 

*■ .(alRlJ3)*Xa'lSt0') = ( a '\B) n(R{ /3)n(a' J S)n(Sf f3') 
03513-14] ={(anaf) J \(RnS)}n{(RnS)[^nP)} 

[*35'11] ={(a«a , )'J( J Rn>Sf)f(/3n / S')}.Dl-.Prop 

*3516. . h . (a 1 R) n S = a \ (R n 8) ^Rna^S [Similar proof to *351 3] 

*3517. b.tR\-/3)AS**(R*S)t/3 = Rn8f.l3 [Similar proof to *35-l 3] 

*3518. +.(cL J \Rfl3)*S = a J \(RnS)tl3-=Rn<z'\Stl3 

[Similar proof to #351 5] 
*3521. h.«1£r0=(a1#)f£ = «1W£) 
Dem. 

Y . *35-102 . D Y : x (a'1 JRf j8) y.=.xea. xRy . y e £ . 
[*351] = . « (a 1 E) y . y € /3 . 

1*35-101] =..»{(*.1J2)r/8)y (1) 

h- . *35102 . D J- : x {a\ R\:@)y . = . x ea . xRy . y e j3 . 
[*35101] =.j.e«. x(R 1/3) y . 

j>35-l] =.x{aMRtl3)}y (2) 

H,(l).(2).Dh.Prop 

*3S22. h.(a^)|^ = a |(S|fi') 
Dem. 

r- . *34.l . D h :, x {(a 1 i2) | S] y . = : <gs) . .a? (a 1 i2) z . s% : 

[*35'1] = : faz) .x e a .xRz . zSy: 

[#10*35] = : area r (gs) .xRz .zSy . 

[*34-l] = :xea.x(R\S)y.: 

[¥351] = : :a?{«1 (R \ S)\y:. >f .Prop 

*35 23. I-. . flf | (Rf &y=.{S ■[.££, [Similar proof to *35-22] 

*3524 a 4 \R\S^(<i J \R)\8 Df 

*35^25. S\Rl@=(S\M)fP Df 

*35'26. >-.(o1^)|(«f/9>*o1(fi|i8)f: i 8.= ^1(i2|S)^/8 = o'1{(Sfi8f)^} 

H(*1*)W£««0'fSW£)} 
= (a]R j £)f £ -^l (R \.S [-ft 
Dem. 

K*341 . Dfer :. ■« {(ali2) |,(flr#)}.y . = : (a*) . »<p%R)z .z(S[0)y : 
[*35'1101] = : (g*) .xea. xRz . z8y . y e/3 : 

'■{#10*35] = : x ea.y e./3 •■: (g^) . a?jRs .zSy : 

1*34-1] => : a; e a . x (R j S)y .ye/3: 

[*35102] =:x\a\(R\£}fft\y (1) 

h . (1) . *36-21-22-23 .{#35 24-25) . O h .Prop 



SECTION D] LIMITED DOMAINS AND CONVERSE DOMAINS 269 

#3527. a.TJ2|#r£-=(«1.R|j8f)r£ Df 
#35 31. h.(R[a)[0 = Rf(otn0) 
Bern. 

r . #35101 . 3 h : x {(R^a) t P) V • = ■ ^W«)y -V e ft • 
[#35-101] =.xRy.yea.ye0. 

[#22-33] = . xRy .yearxft. 

[#35-101] = .x{Rf (a n &)} y : 3 > . Prop 

#35-32. \-.a'\(/3 J \R) = (ar\l3) J \R [Proof similar to that of #35-31] 

#35-33. \-.(a J \R[0)[y^{a d \R[(J3ny)} [ Proof similar to that of #3531] 

#35-34. \-.a J \(/3 J \R[y)~{(an/3) J \R[y} [Proof similar to that of #35-31] 

#35-35. k . «\R = (an D f E) 1 22 

Z)em. 

h . #35-1 .3r-:#(a1.ft)2/. = .#ea. ^ify . 

[#33-14] = .#ea.a:eD<i2.a;%. 

[*2£-33k*35'l J =,*{(an D'R) J \R}y:Db. Prop 

#35-351. f . R T /3 = R [ (0 n (F.K) [Proof as in #35-35] 

#35-352. h . a 1 E f £ = (an B'R) 1 R [ (J3 n d'R) [Proof as in #3535] 
#35-354. \-.(R[a)\8 = R\a J \S 

Dem. 

Y . #341 . #35101 . 3 

V : x {(R f a) 1 8} z . - . fay) . xRy .yea.ySz.. 

[#35-1] = .(<&y).xRy.y(a J \S)z. 

[#341} = . x \R\ (a 1 £)} z : 3 r . Prop 

#35-41. K(auo01.R = a1.R»a'122 [#35'1 . #2234] 

#35412. \-.R[(0yj0') = R[0KjR[0' [#35-101 . *22;34] 

#35-413. h.(«u a')1 £ £(0 u £') = (« 1 £ .££) .o(a1.i2 f^') 

u (a 1 i2 f /3) (a |- JB f £*) [#35102 . *22'34] 
#3542. r-.«1(iJafif) = (o1i2)c»(a1flf) [#351 . #23-34,] 

#35-421. h . (R a £) f/3 = (R f/S) o (£ f £) [#35-101 . #23*34] 

#35-422: b .n^Rv S)f/3 = (aj R\-/3)w(a1 S {&) [#35102 . #2334] 
#35-43. J-:a<:^.D.«1i2G/31i2 
Dem. 

r . #35-1 . 3 h :. a C /3 . 3 : #(a1 R) y . = . x e a . #ity . 
[#22-1] y.xe/3.xRy. 

[*35-l] D.^(/S1i2)2/:.Dr.Prop 

#35-431. Y-.ftCy.l.RTPQRty [Proof similar to that of #35-43] 
#35-432. r-:aC 7 ./3CS. 3. a1i2^G y^M^S 

[Proof similar to that of #3543] 



270 MATHEMATICAL LOGIC [PART I 

*3544. \-.a1RGR 

Dem. 

h . *35*1 . D I- : x (a ^ R) y . D . x e a . xRy . 

■[#3*27] D . xRy : D K. Prop 

*35 441. h.RffiCZR [Proof similar to that of *35'44] 
*35442. h.alRfflGR [Proof similar to that of *3544] 

*35451. bzV'RCa.D.alR^R 

Dem. 

h . *47l . D I- :. Hp . D : xeV'R . = . xeD'R . xe a : 

[*4-36] D-.xeWR.xRy.^.xelf'R.xRy.xea (1) 

h . *3314 . *471 , Db:xRy. = .xeD'R.xRy (2) 

h.(l).(2).Dh :.Hp. D:xRy. = .xRy.xea. 

[*3&-l] =.ar(o1i2)y:.DI-.Prop 

*35-452. \-:a<RC/3.D.R\-/3 = R [Similar proof] 

*35 453. h : D'R C a . D . a 1 12 T £ = P T/3 [Similar proof] 

*35'454. h : <I<12 C £ . D . «1 12 f/3 = a 1 12 [Similar proof] 

*3546. h:£Gi8f.D.-alJBGo1S 
Dem. 

h . *231 . D h :. Hp, D : xRy . D . xSy : 

[Fact] D : # e a . #12y . D . a? e a . #$y : 

[*351] D :x{a J \R)y. 1 .x(a J [S)y ;.1Y . Prop 

*35-461. hrJBGflf.D.-BriSGiS^ [Similar proof] 

*35'462. b-.RGS.D.alRt&GaYStfl [Similar proof ] 

*35471. I- : <FP n a = A . D . P | (a^j 12) = A 

Dem. 

l-.*34-l.Dh:a;{P|(a > |l2)}^.D.(ay).a ; Py.2/(a1l2)^. 
[*35-l] 3 . (gy) , xPy .yea. yRz . 

[*33'14 . *10\5] D.(>&y).y€<I'P.yea. 

[*22'33 . *24'5] D.g!<PPr>a (1) 

I- . (1) . Transp . *24-51 . D 

I- : a'P n a = A . D . ~ {P | (a 1 12)} z : 
[*ll*ll-3] Dr:a'Pna = A.D.(«, ^).~«{P | (a1 12)}^. 
[*2515] D.P|(a1l2) = A:Dr.Prop 

*35'472. h:D f Pna = A.D.(12I k a)|P = A 

*35'473. h:a f Pno=A.D.P|(a1l2^) = A 

*35'474, r:D'Pn/3 = A.D.(a1l2r/3)|P = A 



SECTION d] LIMITED DOMAINS AND CONVERSE DOMAINS 271 

*35'48. \-:a<PCa.-}.P\(a J \R) = P\R 
Dem. 

K*221. Z>r:.Hp.D:ye(I<P.:Vyea: 

[*4'71] liyea'P.yea.^y.yed'P: 

[*10-311] D:aPy.ye(FP.yea.=,,.a:Py.ye<3'P (1) 

b . *3314 . *4'7l . D h : xPy . y <• <PP . = . xPy (2) 

b. (1) . (2) . D h :. Hp . D : #Py . y e a . =„ . a^Py : 

[*10311] D:xPy.yea.yRz.= y .xPy.yRz: 

[*351] D : xPy .y(a J ]R)z.= v . xPy . yRz : 

[♦10-281] D : (ay) . *Py . y (« 1 P) * . = . (gy) . *Py . yP* : 

[*341] D -. x (P | a |P) * . "= . x (P | P) * :. D I- . Prop 

*35-481. K : D'P C /3 . D . (P f/3) | P = P | P [Similar proof] 

*35'51. h.Cnv'(a'jP)=P( k a 

Dem. 

b . *31131 . D H : a; {Cnv'fa 1 P)} y . = . y (a 1 P) # . 
[*351] =.yea.yRx. 

[*3111] =.xRy.yea. 

[*35101] ^.^(Pp^yOh.Prop 

*35 52. I- . Cnv'(P f/3) = £1 P [Proof similar to that of *35'51] 

*35 53. b . Cnv'(a 1 P f /3) = 1 R [ a [Proof similar to that of *35'51] 

*35 61. r . D'(a1 P) = a n D'P 
Dem. 

b.*3S-lB.3b:.x € T>'(a'\R). = :(Ry).x(ci J \R)y: 
[*35 - l] = : (gy) . x e a . #Py : 

[*1035] =:a?ea:(gy).a;Py: 

[*3313] =:xea.xeV<R: 

[*2233] =:«(on D'P) :.0b. Prop 

*35 62. h:aCD f P.D.D f (a1P) = a [*3561 . *22621] 

*3563. h:D ( PCa. = .a > |P = P 

h . *35-61 .Dh:a^|P = P.D.an D'P = D'P . 

[*22621] D.D'PCa (1) 

h.(l).*35-451.Dh.Prop 

*35'64. r.a'(Pf/3) = £n(I'P [Proof as in *35"61] 

*35641. h : a n D'P = A . D . a 1 P = A [*3561 . *33241] 

*35 642. h : a n d'P = A . D . P fa = A [*3564 . *33241] 

*35 643. r : a n D'P = A.D.a1(Pc//S) = a1S [*3564142] 



272 MATHEMATICAL LOGIC [PART I 

#35 644. r : a r> d'R « A . D . (R v S) f a = S f a [*35'642-421] 
#3565. r : C a*!? . D . <P(.R fyS) = £ [#35-64 . #22-621] 

#35 66. h:(I<RC/3. = .R[/3 = R [Proof as in #35-63] 

#35 671. h . D<(R | S) = D'CR fD'tf) 
Dew. 

\-.*33'l3. 3h:.a;€'D'(R\S). = :(<giy).a;(R\S)y: 
[#341] = : (ay, z) . tfifo . .zSy : 

[#11-23] = : (a*, y) ■ #ik ■ *>% : 

[#1035] =:(a*):adk:( ay ).*,Sfy: 

[#3313] =:(a^).«^.^6D f /Sf: 

[*35'101] • = :(a«).a ! (E| k D'i8f)^: 

[#3313] = : m e &'(R [T>'S) :. D H . Prop 

#35-672. h . <T(E | S) = d'((FJR1 £) [Similar proof] 
#35-68. H:an / 8 = A.D.(a > l J R| < /3) 2 = A 
Dew. 

h . #35-61 64-21 . D r . D'(a^ f 0) C a . <P(a1# [/3) C £ . 
[*22-49.*24-13] DH:an/3 = A.D. D'(a1 22 f£) n d'H R[0) = A. 
[#34-531] D.^iJfjS^AOl-. Prop 

#35-7. H:^{(^r/3yy}. = .y6/9.</»(i2'y) 

This proposition is very often used in the later parts of the work. 
Bern. 

h . #14-21 . D h : <j> {(£ f £)'y} . D . E ! (R [/3)'y . 
[#33-43] D.ye<3'0Rf/3). 

[#35-64] 3.ye/3 (1) 

h.(l).«4-7l.DI-:0{(i2r/8)*y}. = ..ye/8-^{(i2r / 8)'y} (2) 

h . #4-73 . *35-101 . D h :. y 6/3 . 3 : a; (.R ftf) y . = x .xRy: 
[#14-272] D : <£{(£ r£)<y} • s . * (* 'y) (3) 

h . (3) . #5-32 . D r : y ■« £ . </> {(U Wy} ■ = . y « £ . $ (R'y) (4) 

K'(2).(4). DKProp 
#35-71. \-:.yel3.D y .R t y = S<y:D.R |\8 = #f£ 
Dew. 
f .#4-7 . D h :. Hp .D : y e . D y .y e . R'y = S'y : 
[#35-7] D:y«i3.D f .(Wy-(Sf%: 

[#35-64] D : y e (I'(i*t0) v, <!<(£ f/3) . D* . (JB f£)«y = (S r/9)<y : 

[#33-45] Dri^^^^t/S^DI-.Prop 

#35-75. h.Ali2 = i2| k A = A'Jief i 8 = o^pA = A 

Dem. 

h. #35-61. Dh.B ( (A J \R) = A. 

[#33-241] Dr-.A1E = A (1) 



SECTION D] LIMITED DOMAINS AND CONVERSE DOMAINS 273 

K*35-64. 0\-.a<(R[A) = A. 

[*33-241] Dh.i2fA = A (2) 

I- . *35-44121 . D h . A^R [fi G A^| R . 

[(1).*25-13] D\-.a\r[I3 = A (3) 

h . *35-44-21 . D\-.a1R[AdR[A. 

[(2).*2513] DKa1£fA = A (4) 

h.(l).(2).(3).(4).DKProp 
*3576. h.Y^\R = RfY = Y]R[Y = R 
Dem. 

h.*351. 3\-:x(V\R)y. = .xeY.xRy. 
[*24104.*473] = . xRy (1) 

j- . *35101 . D h : x (R [ V) y . ° = . xRy .yeY. 
[*24-104.*4-73] = .xRy (2) 

H . *35102 .3h:x(Y]R\-Y)y. = .x€Y .xRy.yeY. 
[*24104.*4-73] = . xRy (3) 

K(l).(2).(3).DKProp 

The rest of this number, down to *35*93 exclusive, is concerned with af/3, 
except *35-81812. 

*35 81. b:x(a J [Y)y. = .xea [*3ol . *25'104] 
*35-812. \-:x(Y [fi) y. = .y e/3 [*35101 . *25104] 
*35*82. h.at/3 = a-JVp/9 
Bern. 

I- . *3 5 • 1 03 . D I- : # (a f £) y . = . # e a . y e £ . 

[*25'104] = . x € a . xYy . y e # . 

[*35-102] =.tf(a1Vf/3)y:Dh.Prop 

*35822. \-.a J \R\-/3 = Rn(a^l3) 
Bern. 

h . *35102 . D h : a?(o^ 12 f £)y . = . x e a . xRy . y e/3 . 

[*4'3] = . xRy . x e a . y e y8 . 

[*35103] = . x Ry . « (a f £) y . 

[*23-33] = . x {R n (a f £)} y : h . Prop 

*3583. b:T><RCa.a'RC0. = .RGatl3 
Dem. 
H.*3314. D\-:.xRy.'^:xe'D i R.y€a t R: 

[*22-46] . D:I><RCa.a<RCj3.3.xea.y<:/3 (1) 

h . (1) . Comm . D h :. D'ECa . d'R C£ . D : xRy . D . aea .y e/3 . 
[*35-103] D.#(af/3)y (2) 

K #35-103. Db:.RGa / t/3.D:xRy.D Xty .xea.ye0: 
[*33-35351] D:I>'RCa.a*RCl3 (3) 

h.(2).(3). DKProp 

R&W I lg 



274 



MATHEMATICAL LOGIC 



[PART I 



#35-831. K Mat£) = (-at£)o(af--£)ci(-at-/8) 
Dem. 



K #23*35.3 J- ::a>{i(at/8)}y.= 
[#35-103] = 

[*45i] = 



M«t£)y}» 

(xea.y e ft) :. 
\. x^ ea.v .y ~ e ft :. 
[#4*42] = :.#~ea:ye/3.v.y~e/8:.v:.#ea.v.a;~ea:y~e/S:. 
[#4 - 4] = z.x<*->ea.yeft . v .xcsjeCL.y~eft.v.xea.y~eft.v.x~eci.y~eft:. 
[#4-25-31-37] 

= :. x »•> ea . y eft . v . xea. . ys eft . v . x ~ ea . y ~ e/3 :. 
[#22*35] =:.xe — a.yeft.v.xett.y€ — ft.v.xe — a.ye — ft:. 
[#35-103] = :. «(- a t ft)y • v . «(« T -£)y . v . x{- a | - ft)y :. 
[#23-34] = :. x {(-a ^ ft) u (a f - £) vy (- a | -/3)} y :: D h . Prop 
#35-832. h.-^(a'| J K^) = (-at/3)c;(at-/3)a(-at-/3)*y-i- J R 

[#35822-831 . Transp . *23'84] 
#35-834. h.(at/3)n( 7 tS) = (an 7 )t(/SnS) 

Dew. 

h . #35-103 . D 

h :a;{(at ft)^(y^ S)}y . = .xea.yeft.xey.ye8. 
[*22-33.*35-103] = . a {(a n 7 ) | (£ n S)} y : D h . Prop 

h.Cnv<(«t/3) = /3t« [#35-103 . #31-131] 
h:a!/8.D.D'(at/8) = « 



#35 84. 

#35 85. 

Dem 



#3586. 

#3587 

Dem. 



K #35-103. #1 0281. D 

I- :. (ay) . a; (a *|* /8) y . = : (ay) . xea . ye ft : 

[#10*35] = :xea:{^y).yeft: 

[*24'5] =;xea.Q\ft 

h . (1) . #33 13 . #10-35 . D h . Prop 

h : a ! a . D . (F(a ^ft) = ft [Similar proof] 

h:a!(aT/3). = .a!«-3!£ 



(1) 



h . #35-103 . D r- :. a ! (a t /8) • = : (3*. y) -ocea.yeft: 
[#1154] = :(a^).«ea:(ay).ye^: 

[#24-5] =:a'.«.a'-/3:.DI-.Prop 

#35-88. h:.at/3 = A. = :a = A.v.£ = A 
[#35-87 . Transp . #2451 . *25'51] 

#35-881. h:<I< J ftCa.D.#|(at£) = D<#t/ 3 
Dem. 

V . #341 . #35103 . D 

h : # {J2 | (a f £)} y . == . (a*) ■ #-R* • ^ea . yeft 



(1) 



SECTION D] 



LIMITED DOMAINS AND CONVERSE DOMAINS 



275 



K*3314. 3\-:.a<RQoL.D:xRz.D.zea: 

[*4-73] D:xRz. = .xRz'.z€a (2) 

\-.(l).(2).D\-::R V .D:.x{R\( a ^/3)}y. = :(' K z).xRz.ye0: 

[♦10'35] = : (rz) . xRz : ye/3 : 

[♦3313] = :xeT>'R.yej3: 

[♦35103] = : x (D'R J/3)y::D\-. Prop 

♦35 882. h : T>'R C £ . D . (a \ /3) \ R == a f <3<£ [Similar proof] 

♦3589. f-:a!/3.D.(at/3)|(/8t7) = («t7):~a!^-3.(«T/3)!(^t7) = A 
Dew. 

I- . *34-l . D h :. x {(a | /3) | (£ f 7 ) } *. 

[♦35*103] = : (gy) .xect.yefi. ye ft . ,Z€ 7 : 

[♦4*24] = :(-jy).#ea. ye/3 . se 7 : 

[♦1035] = : g !/3: xea.zey : 

[♦35'103] = : a !/3:#(at 7 )z (1) 

h.(l).Dh:: a i^.D:*{(ati8)|(/8T7)}'. = -*(at7)*» 

->(a!/3).D:~[^{(at y 8);( / St7)l^"^l-.Prop 

♦35891. H:. a !y8.v.~ a !a: D.(«t/S)!(^ta) = (ata) 
Dem. 

V . #35-88 .Dh:~a!a-^-aT« = A.ati8 = A. 
[♦34-32] D . a *f a = A . (a | £) | (£ t «) = A . 

[♦21-24] D. (a T«) = («t £)!(£?«) (1) 

H.(l).*35-89.Dh.Prop 

a 



♦35-892. h : (a I a) 2 = (a I a) 



♦35-891 



£ 



♦35-895. ,f : a n £ = A . D . (a t /3) 2 = A [♦35-68-82] 

♦359. KD'(ata) = <I<(af a) = C'(af a) = a 
Dem. 

r . ♦35-85-86 . D t- : 3 I a • 3 ■ D'(a| a) = a . <P(a f a) = a (1) 

K*35'88. DH:~a!a.D.~a!(afa). 

[♦3329] D . D<(a | a) = A . <I'(a f a) = A . 

[♦24-51] D . D'(a j a) = a . d<(a f a) = a (2) 

I- . (1) . (2) . *4-83 . D r- . D'(a f a) = <P(a f a) = a . D h . Prop 

♦35-91. hzRGafa.^.C'RCa 

Dem. 

r . ♦35103 .Dh:.22Gafa.=: zrEy . D^j, .x,yea; 
[♦33-352] =:C'RCa:.3h.Yrop 

♦35-92. h :. (ga) . P = f a . D : R G P . = . C'R C CP [*35 9 91] 

18-2 



276 MATHEMATICAL LOGIC [PART I 

*35 93. h : (R) . <f> (D'R) . = . (a) . <£a 

Dem. 

h . *3312 . *1418 . 3 h : (a) . if>a . D . <f> (D'R) : 
[♦10-11-21] Dt-:(a).<f>z.D.(R).<f> (D'R) (1) 

h . *101 . D h : (22) . (D'i2) . D . <f> {D'(a f a)} . 

[*35-9] D.<^a: 

[♦10-11-21] Dh:(iJ).^(D'i2).D.(o).^a (2) 

K(l).(2). DKProp 

*35 931. b:(R).<f>(<I'R). = .(a).<]>a [Proof as in *35'93] 
*35 932. b:(R).<f>(C'R). = .(a).<f>a [Proof as in *35"93] 
*35-94. h:('3_R).<f>(D'R). = .(^a).<f>a [*3593 . Transp] 
*35'941. h : (&R) . <f> (d'R) . = . (ga) . <£a [*35931 . Transp] 
*35-942. r : (gi2) . (CiZ) . = . (get) . <£a [*35"932 . Transp] 



*36. RELATIONS WITH LIMITED FIELDS 

Summary o/#36. 

In this number we are concerned with the special case in which the same 
limitation is imposed \ipon the domain and the converse domain of a relation. 
In this case, the same result is achieved by imposing the limitation on the 
field. It is convenient to be able to regard a "J P[a as a descriptive function 
of a or of P, which we secure by the notation P £ a, whence, as will be ex- 
plained in #38, P fa and £ VP will both mean P £ a. If P is a serial relation, 
and aCC'P, "P£a" will stand for "the terms of a arranged in the order 
determined by P," or, as we may call it briefly, "a in the P-order." P £ a is 
defined as follows: 
#3601. Pta = a J \P[a Df 

We thus have 
#3613. h : x (P I a) y . = . x, y e a . xPy 

Most of the propositions concerning P £ a demand that P should have 
some at least of the characteristics of a serial relation. Hence the propositions 
concerning P £ a which can be given in the present number are, for the most 
part, not the most useful propositions concerning P £ a. The most useful 
propositions in the present number are the following: 

#3625. r:C"PCa.= .Pta = P 

#3629. KP£« = Pnata 

#363. b.Pta = Pt(anC'P) 

#3633. KPtC'P = P 



#3601. P£a = a1Pr« Df 

#3611. h.Pta = a J \P[a [(#36*01)] 

#3613. h:#(Pfc «)#. = .#, yea.xPy [#361 1 . #35102] 

The following propositions are obtained from those of #35 by means of 
#36-11, which, as it is used in each case, is not referred to again. 

#36-2. \-.Pta*Qtft = (P*Q)t(«"/3) [#3515] 

#36201. h.Pta*PtP = Pt(anl3) [*36*2] 

#36-202. h.PtanQta = (PAQ)ta [#362] 

#36203. \-.PtarxQ = (P*Q)ta [#3518] 

#36-21. r . (P I a) £ £ = P fc (a n 0) [*35-33'34] 



278 MATHEMATICAL LOGIC [PART I 

*36-22. h . (P t a) j (Q £ a) G (P | Q) t a 
Dem. 

V . *36'13 . *341 . D V : x {(P £ a) j (Q £ a)} z . = . (gy) .x,y,zea. xPy . yQz . 

[*10-5] D . ( a2 /) .x,zea.xPy. yQz (1) 

I- . (1) . *10-35 . *341 . D h . Prop 
*3623. V.(PvQ)ta = PtoLvQla [*35"422] 
*3624. h«C/3.D.P^«GP^ [*35-432] 
*36 241. h-.PdQ.D.PladQta [*35'462] 
*3625. h:C"PCa. = .P£a = P 

Dem. 

h . *3613 . *4*7 . D h :. P £ a = P . = : #Pt/ . D x<y . x, y e a : 
[*33'352] =:(7<PCa:.DKPrjpp 

*36 26. h : OP n a = A . D . P | (Q £ a) = A . (Q fc a) | P = A [*35-473-474] 

*3627. h:P£A = A [*3575] 

*36-28. h.P£V = P [*3576] 

*3629. h.P^a = Pnat« [*35-822] 

*36 3. \-.Pta = Pt(anC'P) 



Dem. 



h . *3317 . *4-71 . D h : xPy . = .x,yeC ( P. xPy : 

[Fact] D. I- : x, y e a . xPy . = . x, y e a . x, y e (7'P . xPy . 

[*22-33] = .x,yeanC'P.xPy. 

(1) 





[*36-13] = . x [P I (a n C'P)} : y 




h . (1) . *3613 . D h . Prop 


*36-31. 


H:anC<P = A.D.P£a = A [*36-3'27] 


*3632. 


ha rt C'P = i 8nC"P.D.P^ = Pt^ [#36-3] 


*3633. 


Y.PiC'P = P [*36-25] 


*3634. 


1- . Cnv'P I a = (P) £ a [*35'53] 


*3635. 


h.(Pta) 2 G(P 2 )£a [*36-22] 


*364. 


1- : . a n D'P = A . v . a n a*/? = A:D.(Pc/ £)£« = ££« 


Dem, 





r- . *35-643 . D r : a n D'P = A.."} .a\(Rw 8) = a J \S . 

[*35'21] D.(RvS)ta = Sta (1) 

Similarly H : an d'R = A . D . (P vy 5) fc a= 5 £ a (2) 

K(1).(2).DI- . Prop 



*37. PLURAL DESCRIPTIVE FUNCTIONS 

Summary o/*37. 

In this number, we introduce what may be regarded as the plural of R'y. 
"R'y" was defined to mean "the term which has the relation R to y." We 
now introduce the notation "R"ft" to mean "the terms which have the 
relation R to members of ft." Thus if ft is the class of great men, and R is 
the relation of wife to husband, R"ft will mean "wives of great men." If 
ft is the class of fractions of the form 1 — 1 /2 n for integral values of n, and R 
is the relation "less than," R"ft will be the class of fractions each of which is 
less than some member of this class of fractions, i.e. R"ft will be the class of 
proper fractions. Generally, R"ft is the class of those referents which have 
relata that are members of ft. 

We require also a notation for the relation of R"ft to ft. This relation 
we will call R e . Thus R t is the relation which holds between two classes 
a and ft when a consists of all terms which have the relation R to some 
member of ft. 

A specially important case arises when R'y always exists if y e ft. In this 
case, R"ft is the class of all terms of the form R'y when ye ft. We will 
denote the hypothesis that R'y always exists if y e ft by the notation E !! R"ft, 
meaning "the R's of ft's exist." 

The definitions are as follows: 

*3701. R«ft=&{{'&y).yeft.xRy} Df 

*3702. R e = aft(a = R"ft) Df 

*3703. R e = Cnw'(R e ) Df 

This definition serves merely for the avoidance of brackets. Without it, 

"R e " would be ambiguous as between (R) e and Cnv'(.R e ), which are not equal. 
In all cases in which a suffix occurs, we shall adopt the same convention, i.e. 
we shall always put 

^suffix = Cnv'(i2 8uffix ). 
*3704. R'"k = R e "K Df 

Thus R'"k consists of all classes which have the relation R e to some 
member of k. R"'k is only significant when k is a class of classes relatively 
to members of the converse domain of R ; in this case, R'"k is a class of classes 
relatively to members of the domain of R. 

*37'05. Ell R"ft. = :y e ft. 3 y . El R'y Df 

Here the symbol "E !! R"ft" must be treated as a whole, i.e. we must not 
regard it as making an assertion about R"ft. If R"ft = a, we must not suppose 



280 MATHEMATICAL LOGIC [PART I 

that we shall be able to put "E !! a," which would be nonsense, just as "E ! x" 
is nonsense even when x = R'y and E ! R'y. 

The notation R"a, introduced in the present number, is extremely useful, 
and embodies a very important idea. Its use is somewhat different according 
to the kind of relation concerned. Consider first the kind of relation which 
leads to a descriptive function, say D. If \ is a class of relations, D"\ is the 
class of the domains of these relations. In this case, D"\ is a class each of 
whose members is of the form D'jR, where R e X. Again, let us denote by 
"xn" the relation of m to mxn; then if we denote by " NG" the class or 
cardinal numbers, xn"JSfC will denote all numbers that result from multi- 
plying a cardinal number by n, i.e. all multiples of n. Thus e.g. x2"NC will 
be the class of even numbers. If R is a correlation between two classes a and 
fi, i.e. a relation such that, if yefi, R'y exists and is a member of a, while 

conversely, if x e a, R'x exists and is a member of 0, then a = R"/3, and we 
may regard R as a transformation applied to each member of /3 and giving 
rise to a member of a. It is by means of such transformations that two classes 
are shown to be similar, i.e. to have the same (cardinal) number of terms. 

In the case of serial relations, the utility of the notation R"fi is somewhat 
different. Suppose, for example, that R is the relation of less to greater among 
real numbers. Then if is any class of real numbers, R"/3 will be the segment 
of real numbers determined by /3, i.e. the class of real numbers which are less 
than the limit or maximum of /3. In any series, if is a class contained in 
the series and R is the generating relation of the series, R"@ is the segment 
determined by 0. If has either a limit or a maximum, say x, R"fi will be 

R'x. But if has neither a limit nor a maximum, R"/3 will be what we may 
call an "irrational" segment of the series. We shall see at a later stage that 
the real numbers may be identified with the segments of the series of rationals, 
i.e. if R is the relation of less to greater among rationals, the real numbers 
will be all classes such as R"/3, for different values of /3. The real numbers 
which correspond to rationals will be those resulting from a ft which has a 
limit or maximum; the irrationals will be those resulting from a /3 which has 
no limit or maximum. 

The present number may be divided into various sections, as follows: 
(1) First, we have various elementary properties of the terms defined at the 
beginning of the number; this section ends with *37'29. (2) We have next 
a set of propositions dealing with relative products, and with such symbols as 
P"Q"y, P"Q'" K , and so on. The central proposition here is 

*37'33. h . (P | Q)«ry = P"Q"y 

By the definition, Q'"k = Q e " K . Thus P"Q'" K = (P j Q,)«*. This connects 
propositions concerning such symbols as P"Q'" K with propositions concerning 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 281 

relative products. This second section consists of the propositions from #37*3 
to #37 39. (3) We have next a set of propositions on relations with limited 
domains and converse domains. The chief of these are 

*37'401. h.D'(R\-/3) = R"/3 

*37412. h.(R\-a)"l3 = R"(aK/3) 

#3741. I- . T>'(R I a) = a « R"a . <I'(R £ a) = a n R«a 

These propositions on relations with limited domains and converse domains, 
together with certain others naturally connected with them, extend from #37*4 
to #37*52. (4) We next have a number of very important propositions on the 
consequences of the hypothesis E!!i?"/3, i.e. the hypothesis that, for any 
argument which is a member of /3, R gives rise to & descriptive function R l y. 
The chief proposition in this section is 

#37-6. h : E !! R"$ . D , E"£ = x {(gy) .yeP.tc** R'y] 

Propositions with the hypothesis E \l.R"fi are applied to the cases of R 

and R, in which the hypothesis is verified. This section extends from #37*6 
to #37*791. (5) Finally, we have three propositions on the relative product 
of a "f jS with other relations. These propositions are useful in relation- 
arithmetic (Part IV). 

The propositions of the present number which are most used in the sequel, 
apart from those already mentioned, are the following (omitting such as merely 
embody definitions): 

#3715. I- . R"a C D'R 

#3716. h.R"aCa'R 

#372. r:aC/3.D.P"aCP"/3 

#3722. h.P"(au/3) = P"avP"£ 

#3725. I- . D'R = R«a<R . d'R = R"I><R 

#37-26. h . R"/3 = R"(/3 n d'R) 

#37-265. h . R"a = R"(a n C'R) . R"a = R"(a n C'R) 

#3729. h.E"A = A.P"A = A 

#3732. \-.~D<(P\Q) = P"D<Q.a'(P\Q) = Q"(l<P 

#3745. I- :. (y) . E ! R'y . D : g ! R"/3 . = . 3 ! £ 

#37 46. Yixe R"a . = . g ! a n R'x 

#3761. h ::Ell R"t3 .0 :. R"j3 Ca . = :y e j3 .1 y . R'y ea 

For example, let R be the relation of father to son, /3 the class of Etonians, 
a the class of rich men; then "R"@Ca" states "all fathers of Etonians are 
rich," while "yefi.Dy.R'yea" states "if a boy is an Etonian, his father 



282 MATHEMATICAL LOGIC [PART I 

must be rich." In virtue of the above proposition, these two statements are 
equivalent. 

#3762. h:E! R'y .y ea.D .R'ye R"a 

#3763. b::EV.R"a.D:.xeR"a.D x .>lrx: = :y€a.D y .yJr(R'y) 



#3701. R"8 = x{(>&y).ye8.xRy} Df 

#3702. R e = aP(a = R"/3) Df 

#3703. ^ e = Cnv'(i2.) Df 

#3704. R<"K = R e "K Df 

#3705. EUR"8. = :y€8.D y .ElR'y Df 

#371. I- : x eR"8 . = . (ay) .yeS.xRy [#203 . (#37-01)] 

#37101. b:aR e 8. = .a = R"8 [#213 . (#37-02)] 

#37102. h : a (R) e 8 . = . a = R"0 [*37'101] 

#37-103. b : a e R"'k . = . (g£) . 8 e k . a = R"8 . = . a e R e "ic 
[#371-101 . (#37-04)] 

#37-104. I- :. E !! R"S . = : y e £ . D„. E ! R'y [*4-2 . (#37-05)] 

#37-105. b-.xe R"8 . = . (ay) . y e /3 . yito [#37-1 . #31-11] 

#37106. \-:.ElR'x.D:xeR"8. = .R'x€8 
Dem. 

b . #37-105 . #30-4 . D I- :. Hp . D : x e P"/3 . = . (gy) .ye8.y = R t x. 
[#14205] = • R'x e 8 :. D h . Prop 

#3711. h . P e </3 = R"8 [#37-101 . #303] 

#37111. K E! 22/0 [#37-11. #1421] 

#3712. b:(8).R"8 = Q'8. = .R e = Q [#30-42 .#37-11111] 

#3713. \-:P = Q.0.P"8=Q"8 
Dem. 

b . #21-43 . D r- :. Hp . D : #Py .=«,„. a% : 
[Fact] 0:ye8.xPy.= Xi y.ye8.xQy: 

[*10281] 0:(^y).ye8.xPy.^ x .(^y).ye^.xQy: 

[#37-1] D:#6P"£. =s.#eQ"/3:.DKProp 

#37131. h:P = Q.D.P e = <?« 
Dew. 

1- . #37-13 . D h : . Hp . D : a = P"£ . =„ (/3 . a = Q"# : 
[#37-101] D:aP f 8.= a ^.aQ f 8:.Db.Prop 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 283 

*3714. \-:P=Q. = .P ( = Q e 
Dem. 
h. #87101. *21 15. D 
\-:.P e = Q e . =:a = P"(3. = a>fi . a = Q"/3: 
[*13183] = : (£) . P"/3 = Q«&. : 

[*371.*20-15] = : (fr *) : (ay) .ye^.xPy.=. (gy) .yeJB.xQy: 
[#101] D : (a;) : (gy) . y e2(* = w) . #Py . = . (gy) . y ez(z = w) . xQy : 

[*20-3] D : (a;) : (gy) . y = w . aPy . = . (gy) .y = w.xQy: 

[#13-195] D : 0) : xPw . = . xQw (1) 

K(l). #1011-21. #11-2. 1) 
h :. P e = Q € . D : (a?, w) : xPw . .=■ . «Qw : 

[#2143] D:P=Q (2) 

I- . (2) . #37131 . D h . Prop 

#3715. k.P"aCD'P 

Pew. 

h „*371 . D K : a; e 2^'a . D . (gy) -yea. xRy . 

[*10-5] D.(gy).aPy. 

[#3313] D . a e D'P* OK Prop 

#3716. \-.R"aCa ( B [*3715 ^ . #332] 

#3717. \-:.R"0Ca. = :y€&.xRy.D Xiy .xea 
Dem. 

r- . #371 . D h :. P"/3 C a . = : (gy) . y e /3 . xRy . D x . a; e a : 
[*10-23] =:yeJ3 . xRy . "5 x>y . x e a :. D h . Prop 

#37171. H.P^CjS.^tfea.^Py.D^.ye/S 
Pew. 

h . #37105 . D I- :. P"a C £ . = : (gar) .««. aPty . 3, . y e £ : 
[#10-23] = : a? e a . a;Py . D,,,, . y e £ :. D r- . Prop 

#3718. H:ye£.D.P'yCP"/3 
Pew. 

h . #32-18 . D h : . Hp . D : a; e R'y . D . a;Py . y € £ . 
[#37-1] D.a;eP"/3:.DKProp 

#37181. \-:xea.D.R'xCR"a [Proof as in #3718] 

#372. h : a C £ . D . P"« C P"/3 
Pew. 

h . #22-1 . 3 h : . Hp . D : y e a . D y . y e £ : 

[#10-31] Diyea.xPy.Dy.yefi.xPy: 

[#10-28] D:(gy).yea.a;Py.D.(gy).ye/3.a;Py: 

[#37-1] D : a? eP"a.D. x eP"/3:.Dh. Prop 



284 MATHEMATICAL LOGIC [PART I 

The above proposition (#37-2) is one of the forms of asyllogistic inference 
due to Leibniz's teacher Jungius. The instance given by Jungius is: " Circulus 
est figura; ergo qui circulum describit, is figuram describit*." Here the class 
of circles is our a, the class of figures is our £, and the relation of describing 
is our P. 

*37-201. \-:PGQ.D.P"aCQ"a [Similar proof] 

*37202. h:aCyQ.PGQ.D.P"aCQ"/3 [*37-2201] 

*37'21. I- . P"(a n/3)C P"a n P"£ 
Dem. 

b . #371 . D f- :.xeP"(a r\&). = : (33/) . y ea r» £ . xPy : 
[*22-88] = : ( 3 y) . yea . ye/3 . *Py : 

[*10*5] D:(ay>.yea.arPy:(ay).ye£.a>Py: 

[#371] D:#eP"a.#eP",3: 

[#2233] D : x e P"a n P"/3 :. D h . Prop 

#37211. K(PA0"«CP"anQ"o [Similar proof] 

*37212. \-.(PnQ)"(an/3)CP"anP"/3nQ"anQ"l3 [#37-21-211] 

#37*22. KP"(au/3) = P"auP"/3 

This proposition is very frequently used. The fact that here we have 

identity, while in #37*21 we only have inclusion, is due to the fact that 

#1042 states an equivalence, while #10*5 only states an implication. 
Dem. 
b . #37-1 . D h :. x e P"(a «£). = : (ay) .yeaufi. xPy : 
[#22-34] = : (ay) :yea.v.ye/3:xPy: 

[*4-4] = : (ay) : y e a . xPy . v . y e /9 . #Py : 

[#10-42] = :(ay).yea.#Py:v:(ay).ye£.aPy: 

[#37-1] =:*6P"a.v.^P"i9: 

[#22-34] =:#eP"auP" / 8:.DK. Prop 

*37'221. K(PoQ)"a = P"ac-Q"a [Similar proof ] 

#37222. I- . (P a Q)"(a v £) = P"a w P"£ u Q"a u Q"/3 [#37-22-221] 

#37-23. b . D'P e = a {(a/9) . a = P"/3} [*37'101 . #33-1 1] 

#37231. KCFP e = Cls 

The type of "Cls" here is that type whose members are of the same type 

as Q'R. In the proof, use is made of the convention that a Greek letter 

always stands for an expression of the form 2 (<f> ! z). 
Dem. 
b. #37-101 . Db:aR e $(<f>lz). = .a=R" / z(<l>lz): 

[#10-11-281] Db : (a«) . aRJ (cf>lz). = . ( a a) . a = P"2 (4>l*)'. 

[#33131] . Ob:2(<f>lz)ea<R e . = .(>&a).a=:R"2(<f>lz) (1) 

* We quote from Couturat, La Logique de Leibniz, Chapter in, § 15 (p. 75 n.). 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 285 

h . *20-2 . (*3701) . D r : x {fay) .ye$(<j>lz). xRy) = R"z (<f> ! *) : , 
[*1011-24] Dh:(<£):(aa).a = £"£(<£!*) (2) 

K(l).(2).*2'02. Dr:2(<£!*)eCls.D.2(<£!s)e(RR e (3) 

h . *2041 . *2'02 . D\-:2(<f>lz)e<I<R l .D.2(<f>lz)eCte (4) 

h . (3) . (4) . D h . Prop 

As appears in the above proof, it is necessary, when a proposition con- 
taining "Cls" is to be proved, to abandon the notation with Greek letters, and 
revert to the explicit functional notation. 

*37-24. K-aeD'i^.D.oCD'iZ 
Bern. 

f- . *33;13 . *37101 . D I- :: a e D'R t . = :. (ft/3) . a = R"j3 :. 
[*2033.*371] = :.fa/3):xea.= x .fay).ye/3.xRy:. 

[*1161] D:.X€a.D x :fa0,y).yel3.xRy: 

[•11-23] O x :fay^).ye^.xRy: 

[*H-55] 3.:(ay):*i2y:(a^)-y«/8: 

[•10-5] D* : (33/) . ar% : 

[*3313] ZX, : e D'R :.0 K Prop 

*3725. h.D<£ = i^<a<i2.a<# = E"D'i2 

h.*3313. DH:#eD'.R. = .(ay).#.£y. 

[*3314.*4-71] = . ( 3 y) . y e d'R . xRy . 

[#37-1] =.ar Cj R"a'iJ (1) 

r . *33131 . D h : ye d'R . = . (gar) . «% . 

[*33-14.*4-7l] = .fax).xeD ( R.xRy. 

[*37;105] = .yeS"D'i2 (2) 

K(l).(2).I>r.Prop 

*3726. \-.R"l3 = R"(/3na<R) 

Dem. 

h . *371 . D 1- :. x e R"j3 . = : (ay) . y e£ . xRy : 

[*33'14.*4'71] = : (ay) .ye/3.ye d'R . ar% : 

[*22-33] = : (ay) .ye/8n d'fl . #.% : 

[*37'1] = :#ei2"OSr»(FJ2):.DI-.Prop 

*37261. h . R"$ = R"(0 n D'R) [*3726 . *3321] 

*37 262. h : a n (Pi* = /9n d'ii . D . i2"a = #"/3 [*3726] 
*37 263. h : a n D'i2 = £ n D'E . D . R"a = E"/3 [*37261] 
*37'264. h : 3 ! a n fl"/3 . = . fax, y).xea.yej3. xRy . = . E ! j3 n R"a 

Dem. 
b.*22-33.*37-l.D\-:.RlanR"l3. = :fax):xe(i: fay). ye/3. xRy: (1) 



286 



MATHEMATICAL LOGIC 



[PART I 



(5F> y)-xea.yefi.xRy (2) 

(ay) : y e & - (a*) ■«<«■ #-% : 

('Ky)-y€0-yeR"a; 
a!/3nK (3) 



[#11-55] = 

I- . (1) . #11-6 . 3 h :. g ! a n £"£ . = 

[#37-105] = 

[#22-33] = 

r . (2) . (3) . D h . Prop 

#37265. I- . i2"a = P"(a n C'P) . R"a = P"(a n C'P) 
Dem. 

H . #33-161 . #22621 . D I- . d'R = C'P n d'P . 

[#22-481] D K a n <J'i? = a n C'B n CPE 

[#37-262] D h . R"a = P"(a n (T'iJ) 

h . (1) . #33-22 . Dh.Prop 

#37-27. h : <PP C /3 . D . D'P = R"$ [#22621 . #37-25-26] 

#37 271. 1- : D'P C a . D . d'P = R"a [#22-621. *37-25261] 

#37-28. h.R"V = I)<R.R"V = a<R [#37-27-271 .#2411] 

#37 29. I- . P"A = A . P"A = A 

Dem. 

h . #10-5 . D h : (33/) .yeA. xRy . D . (gy) .ye A 
h . (1) . Transp . #24-53 . D h . ~(gy) .yeA. xRy . 



[#371] 
[#24-51] 

R 



h.(2) 



R 



Dh.~ a !£"A. 
Dh.#"A = A 

D I- . P"A = A 



h . (2) . (3) .31-. Prop 

#37-3. I- . {sg'(P i Q)} <z = P"Q^r 

Dem. 

h . *32-2313 . D 

h.{sg<(P|Q)}<z = £{*(PjQ)s} 

[#34-1] = & {(33/) . aPty . 2/^} 

[*32-i8] =&{(w!j)-*Py.y&*) 

[(#37-01 )] = P"Q'z . D h . Prop 

#37*301. h . {gs'(P j Q)] ^ = Q"P~'x [Similar proof] 

#37-302. h : R = P | Q . 3 . ~R'z = P"^ . P<# = Q"P<tf 

[*373 301 .#3223 23116] 

#37-31. Ksg<(PjQ) = P € j"Q 
Pern. 

t- . #37-11-3 . D h . (z) . {sg'(P I Q)}'« = P e 'Q<s 
h.(l). #34-42. Dh.Prop 



(1) 



(1) 

(2) 
(3) 



(1) 



SECTION D] 



PLURAL DESCRIPTIVE FUNCTIONS 



287 



*37-311. I- . gs'(P | Q) = (Q) e | P [Similar proof] 

#3732. I- . D'(P J Q) = P"D'Q . d'(P j Q) = Q"(I<P 

Dem. 

K #331 3. #341. D 

l-:.*eD'(P|Q)- = :( a *):(ay).*Py.yQ*: 
[#11-23] = : (gy) : (a*) . «?Py ■ yQ^ : 

[#11-55] = : (ay) : ncPy : (a*) . yQz : 

[#33-13] =:(ay).^Py.yeD^: 

[#371] =:a?eP"D'Q 

h.(l). #10-11. #2043. D 

I- . D'(P | Q) = P"D'Q 
H.*33-2. Dh.<J'(PjQ) = D<Cnv<(P|Q) 

[#34-2] = D'(Q|P) 

[(2)] = Q"D'P 

[#33-2] = Q"<FP 

V . (2) . (3) . D I- . Prop 
*37-321. H : d'P C D'Q . D . D'(P j Q) = D'P 
#37-322. H : D'Q C d'P . D . d'(P | Q) = d'Q 
#37323. r- : d'P = D'Q . D . D'(P | Q) = D'P . d'(P | Q) = d'Q 
#37-33. H . (P | Q)" 7 = P"Q" 7 

•r-.*37\L.Dh:.a?e(P|Q)" 7 .= 
[#34-1 .#11 -55] = 



(1) 
(2) 



(3) 

[#37-32-27] 

[#37-32-271] 

[#37-321-322] 



(g«) . * e 7 . a; (P | Q) z : 
(3^ y) • * e 7 . aPy . yQ* : 
(ay, z) . xPy .yQz.zey: 
(ay) : xPy : (g«) .yQz.zey. 
(ay) ■ #Py ■ y e Q" 7 : 
#<rP"Q"7:.DKProp 



[#11-23] 
[#11-55] 
[#37-1] 
[#37-1] 

#37-34. K(P|Q) 6 = P e |Q e 

Dem. 

h . #37-11 . D H . (P | Q)/ 7 = (P | Q)" 7 
[#37-33] = P"Q" 7 

[#37-11] =P.'Q.'y 

h . (1) . #10-11 . #34-42 . D K Prop 

#37-341. h . {Cnv'(P | Q)} e = (Q). | (P). [#34-2 . #37*34] 
#37-35. \-:(z). R'z = P'Q'z . D . (7) . P" 7 = P"Q" 7 
i)ew. 

h . #34-42 .D(-:Hp.D.^ = P|Q. 
[#3713] D . £" 7 = (P I Q)" 7 

[#37-33] = P"Q"7 : D h . Prop 



(1) 



288 



MATHEMATICAL LOGIC 



[PART I 



#37 351. h : (a) . R'a = P'Q"a . D . (*) . R"k = P"Q'"k 
L37-35 Q . #3711 . (#3704)1 

#37 352. h : (a) . P"a = P'Q"a . D . (*) . R'"k = P"Q'"k 
T*37-351 ^ . #3711 . (*37-04)l 

#37 353. h : (z) . P'S'* = P'Q<* . D . (7) . #"#"7 = P"Q" 7 

Dem. 

h . *1*21 . D r- : Hp . D . (*) . E ! -R'S'* . 

[*34-41] D . (*) . P'S'* = (P I £)<* . 

[*14131144] D.(z).(R\ S)<z = P'Q<2 . 

[*37-35] D . (7) • (R I S)"7 = P"Q"7 ■ 

[*37-33] D . (7) . R"S"y = P"Q"7 OK Prop 



#37354. I- : (a) . P'S'a = P'Q"a . D . (*) . P"S"k = P"Q'"/c 
*37-355. V:{z). R'S'z = P"Q<* . D . (7) . P",S"<7 - P'"Q"y 



#37353 ^* 
#37 353 ~ 



#37 36. h . D'P 2 = P"D'P . (FP 2 = P"d'P [#37-32] 
#37-37. H . (R% = (P e ) 2 .[#37-34] 

#37-371. P e 2 = (P«) 2 Df 

This definition serves merely for the avoidance of brackets. Like #37 03, 
this definition will be extended to all suffixes. 

#37-38. H . R 2 'x = R"R l x [*37 -3] 
#37-39. KP 2 "a = P"P"a [#37 33] 

#37-4. Kd'(a > |P) = P"a 
Dem. 

V . #33-131 . #35-1 . b : y e d'(a1 P) . = . (gar) . a? e a . arPy . 

[#37-105] = • y e P"a : D h . Prop 

#37-401. H.D f (PP/S) = P"/3 [Similar proof] 

#37-402. KD'(a1 Pf/3) = a a P"/S . (T(a 1 P T /3) = £ n P"a 

Pew. 

h. #3313. #35-102.3 

h i.iceD^a^Pt/S) . = : (ay) .xea.xRy .yep : 

[#10-35] = :xea:('3y).xRy .ye fiz 

[#37-1] =:xea.xeR"/3: 

[#22-33] ee :#ear»P"/3 (1) 

Similarly 

h : y e (I'(a 1 P f £) ■ = . y e /3 n'P"a (2) 

K(l).(2).DKProp 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 289 

*3741. KD'(i£ta) = anfl"a.a<(E£a) = ani£"a [#37-402. #36-11] 
#37-411. h . (a. 1 R)"0 = D'(a 1 R \ 0) = a n £"£ 

H . #37-401 . D H . (a 1.R)"£ = D'(a 1 R) [ 
[#35-21] =D'(a^| J R| k /3) (1) 

K(l). #37-402. DK Prop 
#37-412. \-.(R[a)"0 = R"(an0) 
Dem. 

l-.*37-401.Df-.(i2| k a)"/8 = D'(i2| k a)f k /8 
[#35-31] "D'JK^on/S) 

[#37-401] «JR"(an£).DKProp 

#37413. KtBfca)"£ = a«.R"(on£) 
Item. 

1- . #37-411 . #35-21 . D r . (JR £ a)"£ = a n (£ f a )"0 
[#37-412] =«n £"(a n^.Dh. Prop 

#37-42. H:#"£Ca.D.Hi2)"£ = i2"£ [#37-411 . #22621] 
#37-421. H:£Ca.D.(i*ra)"£ = i2"£ [#37-412 . #22-621] 
#37 43. H :. C (I'i£ . D : a ! #"£ . = . a ! 
Dem. 

b . *37-401 .#35-65 . D H :. Hp . D : R"0 = I)'(R [0) .0 = a'(R[0) (1) 
r . (1) . #33-24 . Dh.Prop 

#37-431. h.aCD^.DialK.E.g.'a [Proof as in #37-43] 

#37-44. l-:.a^ = V.D: 3 ! J R"^. = . a !/9 [#37-43 .#2411] 

#37-441 J- :. D'i* = V . D : a ! £"« . = . g ! a [Proof as in #3744] 

#37-45. r : . (y) . E I R'y . D : a ! ]?«£ . = . g ! y9 [#33431 . #3743] 

#37-451. h:.(x).ElR<x.OinlR« a . = .nla [Proof as in #37-45] 

#37-46. h : a e £"«. = . a !«*£<# [#371 . #32181] 

#37461. \-:x~6R«a.= .anR'x = A.. = . 4 R<a;C-a [#3746. #24-311] 

#37-462. f : *~« ^"« . = . « n R'x = A . = . R l x C - « [#37 461 . #32-241] 

#37-47. H : a ! o . = . a ! R«<a . = . a I R'"a 
Dem. 

h . #37-45-111 .Dh:a!«. = .a! R<"* . 

[(#37-04)] =.a!i2'"a (1) 

h.(l) 5 . Dh: a !«. = . a !« (2) 

I- . (1) . (2) . D h . Prop 
r&w r 29 



290 MATHEMATICAL LOGIC [PART I 

*37-5. H : 08) . P"/3 = Q'/3 . D . (*) . P"'* ~ Q"/c 
Dew. 

h.*3712.DH:Hp.D.P e = Q. 
[*8713] Z).P e "* = Q"*;. 

[(*37-04)] D . P"<* = Q"k : D h . Prop 

*37'501. > . /8 a d'P C R"R"$ 

Bern. 
V . #371 . *10-24 . D h : y e/8 . tfPy . D . #eP"/3 : 
[Exp.*1011-21] D I- :. y e £ . D : #% . D* : x e P"£ : 
[#4-7] D:wRy."y x .xRy.xeR"0: 

[*10-28] D:(^x).xRy.D.(^x).xRy.xeR (t ^: 

[*33131.*37105] D : y e d'R .D.ye R"R"j3 (1) 

K (1) . Imp . *2233 . D 

h : ye& a (FP . D .yeR"R"P OK Prop 
*37502. h.aAD'PCP"P"a [Similar proof] 

*37 51. f- : /3 C (FP . = . £ C R"R"p 
Dem. 

b . *37501 . *22621 O H : £ C (I'P . D . /3 C R"R"p (1) 

K*3716. Dh:/3CP"P"#0./3C(I<P (2) 

h.(l).(2). Dh.Prop 

*37'52. h : a C D'R . = . a C P"P"a [Similar proof] 

The following propositions, down to *37*7 exclusive, are concerned with 
the special properties of R"/3 which result from the hypothesis E !! P"/S, de- 
fined in #37 05. The hypothesis E !! P"£ is important, because it has many 
consequences and is satisfied in many cases with which we wish to deal. 

*37-6. h:E!!jB"/8.D.i2"/8 = ^{(ay).ye/8.« = i2'yj 

This proposition is very important, and is used constantly. 
Bern. 

h . *37104 . D f- :: Hp . D :. y e £ . D„ : E ! R ( y : 

[*30-4] ? y :x = R l y. = .xRy:. 

[*5-32] li.ye&.x^R'y.^y.yep.xRy:. 

[*10281] D:.( a y).ye/3.^P'y. = .( a y).ye/3.*Py. 

[*37\L] = .areP"/3 (1) 

I- . (1) . *1011-21 . *2033 . D h . Prop 

*37-601. h : (*) . E ! R'x . D . P"V = £ {(ay) . x = P'y} 
Dewi. 

I- . *202 . *10 1127 . Z> I- :. Hp . D : x e V . D x . E ! R'x : 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 291 

[*37\L04] >: E !! R"Y : 

[«37'6] 3 : R"Y = x {(ay) .y € Y.x= R'y} (1) 

H .*24104 . *4'73 . D h :y e V . x = R'y . = . # = i2'y : 

[*10-11-281] D I- : (gy) m yeY.x = R'y. = . (ay) .x = R'y: 

[•20-15] Dh^{( a y). F V., = ^}=^{( a t/). a; = %j (2) 

K(l).(2). DKProp 

*3761. h::E!!l^</3-3"£"£Ca. = :ye/3.D 1 ,.i2<yea 

h.*37l7. 3\-::R"0Ca. = :.ye/3.xRy.D x>y .xea:. 
[*ll-2'62] = :. y e-fi . D y : xRy . i.^ea (1) 

h . *37104 . D h ::. Hp . D :: y e/3 . D v :. E ! R'y :. 

[*30'33] D y :.R'yea. = :xRy.O x .xea (2) 

h.(L).(2).Dh::Hp.D:. J R'«/3Ca. = :ye/3.D 1 ,.i2'yea::Dh.Prop 
*37*62. h:E!E'y.yea.D. J R'yei2"a 

Dem. 

h . *3033 . D 

I- :: E ! i2'y . D :. R'yeR"a . = : xRy .D x .x € R"a (1) 

r-.*3'2. Dh -..yeot.D :xRy .D .yea.xRy . 

[*10-24.*371] D.xeR"a (2) 

h.(2).*1011-21.D\~:.yea.D:xRy.3 x .xeR"a (3) 

K(l).(3). DKProp 

The above is the type of inference concerning which Jevons says*: 
" I remember the late Prof. De Morgan remarking that all Aristotle's logic 
could not prove that •' Because a horse is an animal, the head of a horse is 
the head of an animal.'" It must be confessed that this was a merit in 
Aristotle's logic, since the proposed inference is fallacious without the added 
premiss " E ! the head of the horse in question." E.g. it does not hold for an 
oyster or a hydra. But with the addition E ! R'y, the above proposition gives 
an important and common type of asyllogistic inference. 

*37 63. h : : E ! ! R"a .Dz.xe R"a . D* . ->/r# : = : y e a . D y . i/r (R'y) 
Dem. 
K*371 . DH ::xeR"a.1> x .ylrx: = :.(^y) .yea.xRy . D x .yjrx:. 
[#1023] =:.yea.xRy.D xy .ylrx:. 

[*ll-2-62] =:.y€a.D y :xRy.D x .yjrx (1) 

f- . *37104 . D I- ::. Hp . D :: y e a . D y :. E ! R'y :. 

[*30-33] y : . yjr (R'y) . = : xRy .3 x .yjrx (2) 

h . (1) . (2) . D h . Prop 
This proposition is very frequently used. 

* Principles of Science, chap. i. (p. 18 of edition of 1887). 

19—2 



292 



MATHEMATICAL LOGIC 



[PART I 



*3764. h :. E !! R"a . D : (ay) .yea.yjr {R'y) . = . (ga?) . x e R"a . yfrx 
Bern. 

h . #3033 . D I- :: Hp . D :. y e a . D : yjr (R'y) . s . (ga?) . ccRy .^x:. 
[#5 - 32] D:.yea.ylr(R t y). = :yea:(Rx).xRy.'tyx (1) 

K(l).*1011-2r.281.D 



(32/) = y e a = (a#) ■ o'Ry • ir x '• 
(a*) s (ay) -yea- «Jty : i** - 

(qx) . x € R"a . yjrx :: D h . Prop 



h : : Hp . D : . (gy) .yea.yjr (R'y) . = 
[*11'6] = 

[#37-1] = 

#3765. h:E!!i^'£.aCi2"/3.D.a = P"(#"an/3) 
Dem. 
h . #30-21 . #3-27 . D K :: Hp . D :. y e /3 . D„ : sPy . #% . D . * = as (1) 

K*37l.Dh:. Hp.D: 

x € R"(R"a n £) . = . (ay) .yeR"a n £ . #Py . 



(ay, #) • z € a . zRy .yefi. xRy . 
(ay, z). z ea. zRy . y e /3 . xRy ,z = x 
(ay, z) . zea.y e ft . xRy . z = x . 
(ay).a?.ea.ye£.a?2fy. 
xea.xe R"/3 . 
#ea:. D K Prop 



[*37105.*ll-55] 
[(1).*4-71] 
[#13-194] 
[#13-195] 
[*10'35.*37-1J 
[*47l.Hp] 

#37-66. H:.E!!E"/3.D:aCi?"/3. = .(a7).7C/3.a = ie"7 
Dem. 
I- . #37-65 . fexp . #13195 . #22-43 . D 

h:.Hp.D;aCJ?"/3.D.(a7).7C/3.a= J K" 7 (1) 
h. #37-2 . #1313 . D I- : 7 C /3 . a = P" 7 . D . a C P"/3 : 
[#10-11-23] Dh:(<av).yCl3.a = R"y.D.aCR"/3 (2) 

K(l).(2). Dh.Prop 

#37-67. h :. * e 7 . D* . E ! P'S'^ : D . R"S"y = & {(a*) .zey.x = R'S'z] 
Dem. 

h. #34-41. D\-:H.p.zey.D z .R'S'z = (R\S)'z (1) 

h . (1) . #14-21 . D h : Hp . zey . D z . E ! (R \ S)'z (2) 

K(2).*37-6. 0\-:U^.O.(R\S)"y = x{(^z).Z€y.x^(R\S)'y} 
[(1)] -ft{( a 5).ire 7 .«-iJ'S' 7 } (3) 

h. #37-33. D\-.R"S"y = (R\S)"y (4) 

K(3).(4). DKProp 

#3768. hz.zey.D,. P'Q'z =R'z:D. P"Q"y = R"y 
Dem. 

V . #14-21 . D h : Hp . z e 7 . D . E ! P< Q's . E ! R l z . 

[#34-41] D . P'Q'z = (P I Q)'« . E ! R'z . (1) 

[*14-21-131144.Hp] D.E!(P|Q)'*.(P|Q)'s = P's (2) 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 293 

f-.*37-33.Dh.P"Q" 7 = (P|Q)"7 (3) 

K(2).(3).*37'6.D 

l-:Hp.D.P"Q"7«a{(a*).*€ 7 .a>«(P|Q)<*} 

[(2)] = x{(nz).zey.x = R<z\ 

[#37-6.(1)] = R"z : D H . Prop 

#3769. h :. y € 13 . D y . R'y = S'y : D . R"$ = £"/9 

Dem. 

h. #14-21. Dh::Hp.D:.yei8.D.E!i2'y.E!S'y:. (1) 

[#30-4] 3:.ye@.D:xRy. = .x = R'y. 

[#14-142] = .x = S'y. 

[#30-4.(1)] = .xSy:. 

[#5-32] D:.ye/3.xRy. = .yel3.xSy (2) 

H . (2) . #1011-21-281 . D 

I- :. Hp . D : (ay) . y € £ . a;% . = . (gy) .yefi.xSy: 

[#37-1] D:a:eP"/3.= .#e£"£:.DI-.Prop 

— > «— 

A specially important case of iJ"/3 is i2"y8 or R"f3. This case will be 

further studied later (in #70); for the present, we shall only give a few 

preliminary propositions about it. It will be observed that the hypothesis 

E !! P"£ or E !! P"£ is always verified, in virtue of #3212121. Hence the 
following applications of #376 ff.: 

#377. KJR"£ = a {( a y).y ej3.a = R f y) [#376 . #3212] 

#37-701. \-.*R"a = l3{(>&x).x€a./3 = R'x} [#37-6 . #32-121] 

#37-702. t-:.R"j3CK. = :ye/3.3 y .R'yeK [#37-61] 

#37703. Y'.. < R"PQic. = ixeft>'} x .R l xeK [#3761] 

#37704. h:yea.:>.P<yeP"a [#37 62 .#3212] 

#37-705. \-:xea.D.*R'x€*R"a [#3762 .#32121] 

#37-706. h : . a e~R"& .D a .fa: = :ye^.D y .^ {R'y) [#37-63] 

#37707. h : . /3 e*R"a . D fi . yjr& : = : x e a . D x . yfr (R'x) [*37 63] 

#37-708. h : . (got) . a e £"/3 .yfra. = . (ay) -yefi.ty (R'y) [*37'64] 

#37-709. h : . (go) . a e R"p . i/r« . = . (g*) .xe$.y\r (%'x) [*37 64] 

#37-71. H : * C 5"/9 . D . « = R«{(Gnv*R)"K « £} [*37'65] 

#37-711. h:ArCP"/9.D.A; = ir"{(Cnv'S)"/cn / 8} [*37'65] 

#37712. f-:«CP"/3. = .(a7).7C^.«: = i2"7 [#3766] 

#37-713. \-zkC P"/3 . = . (37) . 7 C £ . * = R*'y [#37-66] 



294 MATHEMATICAL LOGIC [PART I 

#3772. h:£ = PjQ.D.?" 7 = P"<"<2" 7 
Dem. 

h . #37-1 1-302 . D h : Hp . D . (z) . P e <~Q ( z = R'z . 

[*37'68] D . P € "Q" 7 = P" 7 . 

[(♦37-04)] D . P'"~Q"y =^fl" 7 : O h . Prop 

♦37721. h : R = P | Q . D . P" 7 = Q' "P" 7 [Proof as in ♦3772] 

♦37-73. h:^l/3. = .'3_lR tt /3. = .'^lR (t /3 [♦3745 .♦3212121] 

♦37-731. H:y9 = A. = .P" y S = A. = .P" / 8 = A [♦37 73 . Transp] 

Observe that the A's which occur in this proposition will not be all of the 
same type. E.g. if R relates individuals to individuals, the first A will be 
the class of no individuals, while the second and third will be the class of 
no classes. Thus the ambiguity which attaches to the type of A must be 
differently determined for different occurrences of A in this proposition. In 
general, when this is the case with our ambiguous symbols, we shall adopt a 
notation which indicates the fact. But when the ambiguous symbol is A, it 
seems hardly worth while. 

♦3774. \-:./3Ca'R.= :aeM"/3.D a .'3_lcL 
Dem. 

h . ♦37-706 . D h '..aeR"P . D a • 3 ! « : = : ye& . D„ ■ a \R'y : 
[♦3331] = : C d'R :. D h . Prop 

♦37-75. \-:.aCT>'R. = :/3eR"a.'Df i .<&l/3 [Proof as in #37 -74] 

#37-76. KP'MSCCls 
Dem. 

h . #37-7 . D h :. a e R"& . D : (ay) .yefi.ct = R'y: 

[♦10-5] D: ( 3 y). a = £<</: 

[♦32-13] D : (gy) . a = & (xRy) : 

[♦20-16] D:(a<£). « = £(</>!#): 

[♦20-4] D : a e Cls : . >h . Prop 

♦37-761. h.^'aCCls [Proof as in ♦37 76] 

♦37-77. h : a e R"<I'R . D tt . a ! « [#3774 . #22 42] 

♦37771. h : /3 e K"D<£ . D p . a ! /8 [Proof as in #37-77] 

♦37-772. KA~eP"(TP [♦3777 . ♦2463] 

♦37*773. h.A~eP"D'P [*37'771 . #2463] 

#37-78. r . D'~R =~ft"V [#3728] 



SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 295 

*37'781. )r,D'R = R"Y [*3728] 

*37 79. h . E"V = o {(gy) . a = R'y} [*37601 . *3212] 

*37791. H . R"V = $ {(ga?) . £ = jR'a?} [*37-601 . *32 121] 

*37'8. h.(at/3)|/S = atS"^ 
Dera. 
h . *35103 . *341 .3\r:x{(a\0)\S}e.=>. (gy) .xea.yefi.ySz. 

[*10-35.*37-105] =.xea.z e 5"£ . 

[*35103] = .#(af S"/8)*:DI-.Prop 

*37 81. I- . i2 1 (a t £) = (#"a) t £ [ Proof as in *37"8] 

*37 82. h . iZ | (d t £) | iS = ( #"a) t (#"£) [*37'8-81] 



*38. RELATIONS AND CLASSES DERIVED FROM A DOUBLE 
DESCRIPTIVE FUNCTION 

Summary of #38. 

A double descriptive function is a non-propositional function of two 
arguments, such as ar\/3,ayj fi, RnS, Rv S, R\S, a 1 12, R [a, 12 £ a. The 
propositions of the present number apply to all such functions, assuming the 
notation to be (as in the above instances) a functional sign placed between the 
two arguments. In order to deal with all analogous cases at once, we shall in 
this number adopt the notation 

where "?" stands for any such sign as n, v, f\, vy, |, \, \, £, or any functional 
sign to be hereafter defined and satisfying the condition 

(x,y).Kl(x%y). 
The derived relations and classes with which we shall be concerned may be 
illustrated by taking the case of a r\ jS. The relation of a r\ yS to /3 will be 
written a n, and the relation of a n fi to a will be written n ft. Thus we 

shall have 

Kan/3 = ar>'£ = n£'a. 

The utility of this notation is chiefly due to the possibility of such notations 
as a.t\ if K and n /3"/e. For example, take such a phrase as "the foreign 
members of English Clubs." Then if we put a = foreigners, k = English Clubs, 
we have 

«rt"« = the classes of foreign members of the various English Clubs. 
Or again, let o be a conic, and k a pencil of lines; then 

a n"« = the various pairs of points in which members of k meet a. 
In this case, since a n /3 = ft c\ a, we have a n = n a. But when the function 
concerned is not commutative, this does not hold. Thus for example we do 
not have R\ = \R. 

The notations of this number will be frequently applied hereafter to R | S. 
In accordance with what was said above, we write R \ for the relation of R j S 
to S, and | S for the relation of R | S to R. Hence we have 

12 1 '£ = 1 £'12 = 5 1 & 
Hence | S"\ will be the class of relations obtained by taking members of \ 
and relatively multiplying them by S. Thus if X were the class of relations 
first cousin, second cousin, etc., and S were the relation of parent to child, 
| S"\ would be the class of relations first cousin once removed, second cousin 
once removed, etc., taken in the sense which goes from the older to the younger 
generation. 



SECTION d] operations 297 

It is often convenient to be able to exhibit | S"\ and kindred expressions 
as descriptive functions of the first argument instead of the second. For this 

purpose we put 

\\S=\S"\ 

with similar notations for other descriptive double functions. We then have, 

just as in the case of R I $, 

\\<S=\S'\ = \\S. 
>> >> ft 

This enables us to form the class \|"/t. This class is chiefly useful because 

jj 

the members of its members (i.e. s'\|"/a, as we shall define it in *40) con- 

stitute the class of all products R \ S that can be formed of a member of X and 
a member of fi. 

Thus we are led to three general definitions for descriptive double functions, 
namely (if x ? y be any such function) 

a; $ is the relation of x % y to y for any y, 

-$■ y „ » « »> » x » x > 

a ? y is the class of values of x % y when x is an a. 

Since a % y is again a descriptive double function, the first two of the above 

definitions can be applied to it. The third definition, for typographical reasons, 
cannot be applied conveniently, though theoretically it is of course applicable. 
The relations x % and $ y represent the general idea contained in some of the 
uses in mathematics of the term "operation," e.g. + 1 is the operation of 
adding 1. 

The uses of the notations introduced in the present number occur chiefly 
in arithmetic (Parts III and IV). Few propositions can be given at this stage, 
since most of the important uses of the notation here introduced depend upon 
the substitution of some special function for the general function " °. " here 
used. In the present number, the propositions given are all immediate con- 
sequences of the definitions. 

*38-01. x% = uy(u = x%y) Df 
*38-02. %y = ti&{u, = x%y) Df 
*3803. a?2/ = ?2/"a Df 

*381. Y:u{x%)y. = .u = x%y [(*38-01)] 

*38101. V:u{%y)x. = .u = x%y [(*38'02)] 

*3811. Y .x%'y = %y'x = x%y [*38T101 . *30"3] 

*3812. KE!a>?'y.E!?3f'a? [*3811 .*1421] 

*3813. \--.uex%"a. = .{'&y).yea.u = x%y [*381 .*37l] 

*38131. r- : u e $ y«a . = . (&x) .xea.u=x%y [*38-K)l . *371] 



298 MATHEMATICAL LOGIC [PART I 

*38'2. h.a%y = %y"a. [(*38'03)] 

*38'21. V.a%y = u{('$x).xecL.u = x%y) [*38-2131] 

*38-22. h.a?'y = ?y'« = «¥y [*38-ll] 

a >> it 

*38 23. h . E ! a % 'y . E ! ? y'a [*38'22 . *14*21] 

tt >) 

*38'24. h:a!a?y. = .a!a 

Bern. 

h.*38-2.*37-29.Transp.Dh: a !a$2/.D.a!o (1) 

b.*38-21. H-:x € a.D.(x$y)ea$y. 

[*10-24] D.g!a?y (2) 

I- . (1) . (2) . D h . Prop 

*38'3. Ka?"/3==^{(ay).2/e/3.7 = a?2/}=7{(ay)-2/ e ^. 7 = ?2/"«} 

[*38-13'2] 
*3831. h.?/^ = ^{( a a).ae/ C .7 = a? 2 /} = 7{( a a).a € ^. 7 ==?y"a}==?/"/c 

[*38-131-2.*37103] 



NOTE TO SECTION D 

General Observations on Relations. The notion of "relation" is so general 
that it is important to realize the different sorts of relations to which the 
notations denned in the preceding section may be applied. It often happens 
that a proposition which holds for any relation is only important for relations 
of certain kinds; hence it is desirable that the reader should have in mind 
some of the principal kinds of relations. Of the various uses to which different 
sorts of relations may be put, there are three which are specially important, 
namely (I) to give rise to descriptive functions, (2) to establish correlations 
between different classes, (3) to generate series. Let us consider these in 
succession. 

(1) In order that a relation R may give rise to a descriptive function, 
it must be such that the referent is unique when the relatum is given. 

— ♦ 4— 

Thus, for example, the relations Cnv, R, R, D, Q, C, R e , defined above, 
all give rise to descriptive functions. In general, if R gives rise to a 
descriptive function, there will be a certain class, namely d'R, to which 
the argument of the function must belong in order that the function may 
have a value for that argument. For example, taking the sine as an illustra- 
tion, and writing "sin';?/" instead of "sin y," y must be a number in order 

that sin's/ may exist. Then sin is the relation of y to a; when # = sin'y. If 
we put a = numbers between — 7r/2 and 7r/2, both included, sin f a will be the 
relation of x to y when x = sin';?/ and — 7r/2 ^?/$7r/2. The converse of this 

relation, which is a "] sin, will also give rise to a descriptive function; thus 

(a "J sin)'# = that value of sin -1 x which lies between — tt/2 and 7r/2. This 
illustrates a case which arises very frequently, namely, that a relation R 
does not, as it stands, give rise to a descriptive function, but does do so 
when its domain or converse domain is suitably limited. Thus for example 
the relation "parent" does not give rise to a descriptive function, but does 
do so when its domain is limited to males or limited to females. The relation 
"square root," similarly, gives rise to a descriptive function when its domain 
is limited to positive numbers, or limited to negative numbers. The relation 
"wife" gives rise to a descriptive function when its converse domain is limited 
to Christian men, but not when Mohammedans are included. The domain 
of a relation which gives rise to a descriptive function without limiting its 
domain or converse domain consists of all possible values of the function; the 
converse domain consists of all possible arguments to the function. Again, if 

R gives rise to a descriptive function, R'x will be the class of those arguments 

for which the value of the function is x. Thus sin'# consists of all numbers 



300 



MATHEMATICAL LOGIC [PART I 



whose sine is x, i.e. all values of sin -1 x. Again, sin"a will "be the sines of the 
various members of a. If a is a class of numbers, then, by the notation of #38, 
2 x "a will be the doubles of those numbers, 3 x "a the trebles of them, and 
so on. To take another illustration, let o be a pencil of lines, and let R'oc be 
the intersection of a line x with a given transversal. Then R"a. will be the 
intersections of lines belonging to the pencil with the transversal. 

(2) Relations which establish a correlation between two classes are really 
a particular case of relations giving rise to descriptive functions, namely the 
case in which the converse relation also gives rise to a descriptive function. 
In this case, the relation is "one-one," i.e. given the referent, the relatum is 
determinate, and vice versa. A relation which is to be conceived as a correla- 
tion will generally be denoted by S or T. In such cases, we are as a rule less 
interested in the particular terms x and if for which xRy, than in classes of 
such terms. We generally, in such cases, have some class /3 contained in the 
converse domain of our relation S, and we have a class a such that a = S"ft. 
In this case, the relation S correlates the members of a and the members of 

\y>;..-' 

/3. We shall have also ft = $"a, so that, for such a relation, the correlation is 
reciprocal. Such relations are fundamental in arithmetic, since they are used 
in denning what is meant by saying that two classes (or series) have the same 
cardinal (or ordinal) number of terms. 

(3) Relations which give rise to series will in general be denoted by P 
or Q, and in propositions whose chief importance lies in their application to 
series we shall also, as a rule, denote a variable relation by P or Q. When 

P is used, it may be read as "precedes." Then P may be read "follows," 

P'x may be read " predecessors of x," P'x may be read " followers of x." 
D'P will be all members of the series generated by P except the last (if any), 
d'P will be all members of the series except the first (if any), G f P will be 
all the members of the series. P"a will consist of all terms preceding some 
member of a. Suppose, for example, that our series is the series of real numbers, 
and that a is the class of members of an ascending series x x , x 2 , x 3 , ... x vy .... 
Then P"a will be the segment of the real numbers defined by this series, i.e. 
it will be all the predecessors of the limit of the series. (In the event of the 
series ajj, x 2 , x 3 , .... x v , ... growing without limit, P"a will be the whole series 
of real numbers.) 

It very often happens that a relation has more or less of a serial character, 
without having all the characteristics necessary for generating series. Take, 
for example, the relation of son to father. It is obvious that by means of 
this relation series can be generated which start from any man and end with 
Adam, But these series are not the field of the relation in question ; more- 
over this relation is not transitive, i.e. & son of a son of x is not a son of x. 
If, however, we substitute for " son " the relation " descendant in the direct 



SECTION D] NOTE TO SECTION D 301 

male line " (which can be defined in terms of " son " by the method explained 
in #90 and #91), and if we limit the converse domain of this relation to 
ancestors of x in the direct male line, we obtain a new relation which is 
serial, and has for its field x and all his ancestors in the direct male line. 
Again, one relation may generate a number of series, as for example the 
relation " x is east of y." If x and y are points on the earth's surface, and in 
the eastern hemisphere, this relation generates one series for every parallel 
of latitude. By confining the field of the relation further to one parallel of 
latitude, we obtain a relation which generates a series. (The reason for 
confining x and y to one hemisphere is to insure that the relation shall be 
transitive, since otherwise we might have x east of y and y east of z, but x 
west of z.) 

A relation may have the characteristics of all the three kinds of relations, 
provided we include in the third kind all those which lead to series by some 
such limitations as those just described. For example, the relation + 1, 
i.e. (in virtue of the notation of #38) the relation of x + 1 to x, where x is 
supposed to be a finite cardinal integer, has the characteristics of all three 
kinds of relations. In the first place, it leads to the descriptive function 
(+ !)'#, i.e. x + 1. In the second place, it correlates with any class a of 
numbers the class obtained by adding 1 to each member of a, i.e. (+ l)"a. 
This correlation may be used to prove that the number of finite integers is 
infinite (in one of the two senses of the word "infinite"); for if we take as 
our class a all the natural numbers including 0, the class (+ l)"a consists of 
all the natural numbers except 0, so that the natural numbers can be corre- 
lated with a proper part* of themselves. Again, the relation + 1 may be used, 
like that of father to son, to generate a series, namely the usual series of the 
natural numbers in order of magnitude, in which each has to its immediate 
predecessor the relation +1. Thus this relation partakes of the characteristics 
of all three "kinds of relations. 

* I.e. a part not the whole. On this definition of infinity, see *124. 



SECTION E 

PRODUCTS AND SUMS OF CLASSES 

Summary of Section E. 

In the present section, we make an extension of o r» 0, a w ft, R n S, R \y S. 
Given a class of classes, say k, the product of k (which is denoted by p ( tc) is 
the common part of all the members of k, i.e. the class consisting of those 
terms which belong to every member of k. The definition is 
p l K — ft (a e k . D tt . x € a) Df. 

If k has only two members, a and (3 say, p f /c = a r\ y& If k has three members, 
o, /3, 7, then p'tc — a n /9 n 7 ; and so on. But this process can only be continued 
to a finite number of terms, whereas the definition of p'tc does not require 
that k should be finite. This notion is chiefly important in connection with 
the lower limits of series. For example, let \ be the class of rational numbers 
whose square is greater than 2, and let " xMy " mean "x<y, where x and y 

are rationals." Then if xeXrM'x will be the class of rationals less than x. 

— > — * 

Thus M"\ will be the class of such classes as M l x, where xe\. Thus the 

— > — > ■ 

product of M"\, which we call p l M"\, will be the class of rationals which 

are less than every member of X, i.e. the class of rationals whose squares are 

less than 2. Each member of M"\ is a segment of the series of rationals, and 

p l M"\ is the lower limit of these segments. It is thus that we prove the 
existence of lower limits of series of segments. 

Similarly the sum of a class of classes k is defined as the class., consisting 
of all terms belonging to some member of k ; i.e. 

s i /c = ft {(ga) K a e k . x e a] Df, 
i.e. x belongs to the sum of k if x belongs to some k. This notion plays the 
same part for upper limits of series of segments as p'tc plays for lower limits. 
It has, however, many more other uses than p f K, and is altogether a more im- 
portant conception. Thus in cardinal arithmetic, if no two members of k have 
any term in common, the arithmetical sum. of the numbers of members possessed 
by the various members of k is the number of members possessed by s'/c. 

The product of a class of relations (X say) is the relation which holds 

between x and y when x and y have every relation of the class \. The 

definition is 

p'\ = xy-(Re\.D B . xRy) Df. 

The properties of p'\ are analogous to those of p'tc, but its uses are fewer. 



SECTION El PRODUCTS AND SUMS OF CLASSES 303 

The sum of a class of relations (X say) is the relation which holds between 
x and y whenever there is a relation of the class X which holds between x 
and y. The definition is 

*«X = a0{( a fl).22eX.fl?2fy} Df. 
This conception, though less important than s'/c, is more important than p'\. 
The summation of series and ordinal numbers depends upon it, though the 
connection is less immediate than that of the summation of cardinal numbers 
with s'k. 

Instead of defining p'ic, s'k, p'X, s'\, it would be formally more correct to 
define p, s, p and s, which are the relations giving rise to the above descriptive 
functions. Thus we should have 

p = J3ic{l3 = £(aeic.D a .xea)\ Df, 
whence we should proceed to 

h : @p/c . = . # = £ (a e k . D a . x e a), 

I- .p i K = f&(a.€K .D a .xea), 
and h . E ! p'/c. 

But in cases where the relation, as opposed to the descriptive function, is 
very seldom required, it is simpler and easier to give the definition of the 
descriptive function in the first instance. In such cases, the relation is always 
tacitly assumed to be also defined ; i.e. when we give a definition of the form 

R'x = S'x Df, 
where S is some previously defined relation, we always assume that this 
definition is to be regarded as derived from 

R = ti&(u = S'a:) Df. 

In addition to products and sums, we deal, in the present section, with 
certain properties of the- relations R | and | S, the meanings of which result 
from the notation introduced in #38. Such relations are very useful in 
arithmetic. The reason for dealing with them in the present section is that 
a large proportion of the propositions to be proved involve sums of classes of 
classes or relations. 



*40. PRODUCTS AND SUMS OF CLASSES OF CLASSES 

Summary o/#40. 

In this number, we introduce the two notations (explained above) 

p'tc = & (a e k . D a . x e a) Df 

s'k — ft {(ga) . a e k . x e a} Df 

Both these notations will be found increasingly useful as we proceed, but s'k 

remains more useful than p'ic throughout. It is required for the significance 

of p'tc and s'k that k should be a class of classes. 

In the present number, the most useful propositions are the following : 

*40'12. V : a e k . D . p l K C a 

I.e. the product of k is contained in every member of k. 

*4013. h : a e k . D . a C s'k 

I.e. every member of k is contained in the sum of k. 

*4015. h:./3C j p'K. = :7e«.:V/3C7 

I.e. /3 is contained in the product of k if /8 is contained in every member 
of k, and vice versa. 
*40151. ht.s'KCfl.s'.yeic.Dy.yCp 

I.e. the sum of k is contained in yS if every member of k is contained in ft, 
and vice versa. 
*40'2. I" :* = A.D.p'/e = V 

I.e. the product of the null-class of classes is the universal class. This may 
seem paradoxical at first sight, but it is really not so. The fewer members k 
has, the larger, speaking generally, p l K becomes. If k has no members, then 
k has no members to which a given term x does not belong, and therefore x 
belongs to p l tc. 
#40-23. I- : a ! * . D . p'tc C s'k 

I.e. unless * is null, its product is contained in its sum. 

*40-3& h . B"8'k = s'R'"* 

This proposition is very often used in arithmetic. What it states is as 
follows : Given a class of classes k, take its sum, s'k, and then consider all the 
terms that have the relation R to some member of s'k ; this gives the class 
R"s'k', next, take each separate member of k> say a, and form the class R"a, 
consisting of all terms having the relation R to some member of a. The class 
of all such classes as R"a, for various a's which are members of k, is R"'k ; 
the sum of this class, by the above proposition, is the same as R"s'k. 
*40'4> b -..Ell R"0.1.s<R"@ = ${('Ry).yej3.x€R'y} 

This proposition requires, for significance, that R'y should always be a 



SECTION E] PEODUCTS AND SUMS OP CLASSES OP CLASSES 305 

class. The proposition states that, if R i y always exists when y e /3, then the 
sum of all classes which have the relation R to some member of j3 consists of 
all members of such classes as R'y, where ye/3. 

*40 5. h . s'R"/3 = R"0 

This proposition results from #40'4 by substituting R for R in that 
proposition. 

*40 51. I- . p'R"P = x [y e . D y . xRy) 

In virtue of *40"5, p'R"j3 is correlative to R"ft. Thus if R is a serial 

relation, p'Rt'ft consists of terms preceding the whole of /3, and R lt f3 consists 
of terms preceding part of j3. If has a lower limit, it will be the upper limit or 

maximum of p'R"@; if y3 has an upper limit, it will be the upper limit of R"@. 
#40-61. I- : a ! /3 . D . p<R"/3 C R"p . p'R"/3 C R"/3 

In this proposition the hypothesis is essential, since, if = A, p'R'^ = V 
andi2"£ = A. 



#4001. j»'*=:S(ae«.D a .*««) Df 

#40 02. s'/c = % {(go) .aex.xea} Df 

#401. \-:.xep i K.= :cL€K^ a .xea [*20-3 . (*40'01)] 

«4011. H : x e s'/c . = . (ga) . a e tf . a; e a [*20'3 . (#40'02)] 

#4012. hae*.Dy«Ca 

h.*401 . *101 . Dh:.a;ej//t.D:ae*:.D.a?ea:. 

[Comm] "Zhz.aeic.D-.xep'ic .D.xea (1) 

h . (1) . #1011 21 . *22-l . D r . Prop 

#4013. H:ae*.D.aCs'* 
Z>em. 

h . #4011 . *10-24 .^h-.ae/c.xea.D.xes'ic: 
[Exp] DH :.ae«. D:#ea. "D.xes*/c (1) 

h . (1) . #10-1 1-21 . #221 .0 h . Prop 
#4014. bzaeie.xep'/c.y.xea [*4012 . Imp] 
#40141. tzaeK.xea.D.xes'/c [#4011 . #1 024] 
#4015. h:./3Cp'Ar. = : 7 6/c.D 7 .^C7 
Dem. 

h.*4>Ol.D\-::0Cp t Ki = :.xe0.D x :y€K.D y .X€y:. 
[*11"62] = :. (x, y) : x e f3 . y e k . 2 . x e y :. 

[*4-3'84.*ll-33] =:.(x,y):yeie.xej3.D.X€y:. 

[*ll-2-62] =:.y€K.D y :xe/3.D x .xey:. 

[*22-l] =:.7e*.D y . / SC 7 ::Dh.Prop 

R&w i 20 



306 



MATHEMATICAL LOGIC 



[PART I 



*40151. V;.s ( K C&. = :yeK.O y .yQP 
Bern. 

h . *4011 . D h iis'k C /9 . s :. (37) . 7 e « . x e 7 . D x . x e /3 :. 
[*10*23] = :.(7, x):.y€K.xey.0.xe@:. 

[*ll-62] = :. (7) :.«7 e k . D : (#) : x e 7 . D . a? e /3 :. 

[*22-l] =:.7e/c.D y .7C^::DI-.Prop 

This proposition is frequently used. 
*4016. \-:/cC\.O.p f \Cp'ic 

Dem. 

H . *10*1 . D b :: Hp . D :. 7 e k . D . 7 e \ :. 

[Syll] 0:.7e\.D.a?67:D:76/c.D.a?e7 (1) 

h.(l).*101121.D 

h :: Hp . D :. (y) :. y e\ . "2 . x e y : D z y e k . D . x ey 1. 

[*10-27] D:.(y):ye\.0 .X€y:D:(y):yeK.O .xey:. 

[*40'1] ^-..xep'X.l.xep'ic (2) 

l-.(2).*10-ll-21.DH.Prop 

*40161. h:«C\.3.s'*Cs'X 

Dem. 

\-.*10'l.Db:.B.p.D:y€K.D.yeX: 

[Fact] Ozye/c.scey.'y.'yeX.aey: 

[*10'11 -28] D : (37) . 7 e * . x e 7 . D . (37) . 7 e"X . a; e 7 : 

[*4<011] D-.xes'ie.D.xes'X (I) 

h.(l).*1011'21.Dt-.Prop 

*4017. I-./ku p'X C jp'(/c n \) 

Dem. 

H j*2234.Dh :: x e p* k v p'\ . = :. a?ej9*« . v .xep'X U 

[#401] =:.76«iD Y .a;67:v:7eX.D t .a!67:. 

[*1041] D :. (7) :. 7 e /c . D . a; € 7 : v : 7 e X - D . x e 7 :• 

[*4-79] D:.(y):y€K.ye\.D.xeyi. 

[*2233] D:.(y):y€Kn\.O.xey:. 

[*40-l] D-.-aj^ftX) (1) 

h.(l).*1011.DI-.Prop 

*40171. h.s'«us'X=s'(« uX ) 
Dew. 

I- .*22*34.D h ::«es'« us'\. = wares'*. v.a:es'X. :. 



[*40-ll] 
[*10-42] 
[*4-4] 



• (37) •yeK.xey.M: (37) . 7 e X . x e 7 :. 
. (37) : 7 e /c . x e 7 . v . 7 e \ . x e 7 :. 
■ (27) : * 7€K..v > 7«\:a!€7:. 



SECTION E] 



PRODUCTS AND SUMS OF CLASSES OF CLASSES 



307 



[#22*34] = :.('Ry).y€Ku\.X€y:. 

[#4011] =:.a;e5'(/cw\)::DH.Prop 

#4018. H.jp'(/cw\) = p'/cn^\ 
Dem. 

Y .#401 . D h iixep'iic v \) . = :. y e k v \ . D y . x e y :. 



.(7) z.ye/c .v .y €\:D .xey :. 

. (7) i.ye/c.D.xeyzyeX.D.xeyz. 

• (7) tyeic.D.xey:. (7) iye\ . D . xey :. 

. xep'ic .xep'X :. 

,xep i Kr\p l \ :: D V . Prop 



[#22*34] 
[#4*77] 
[#10*22*221] 
[#401] 

[#22*33] 

#40181. H . s'(/e <\ \) C«"«n«'\ 
Dem. 

h .#40*11 .0 H ::a"€s'(/enX). = :. (g/y) . yexr\ \. xey :. 
[#22*33] =:.(37).7€/e.7e\.a;e7:. 

[#10*5] D :. (*37) iyeK.xey. (37) . 7 e X, . x e y :. 

[#40*11. #22*33] O :. x e s'k n s'\ :: D h . Prop 

#4019. H :: x e «'«■.= :. 7 e k . D y . 7 C /3 : Dp . x e £ 

This proposition is the. extension of #22*6. 

Dem. 

K #40151. D 

b ::y ex .D y .yC -.Dp . x e :. = i.s'tcC .0? .x e (1) 

h . #10*1 . 3 h : .V* C £ . D* . e £ : D : «'* C 0'* . D . e «'« : 



[#22*42] O:0€0'a: 

I- . #22*46 . h :. x e s'* . s'k C £ . D . x e £ :. 
[Exp] >h :.a; e 0'* » D : 0</e C £ . D . a; e :. 

[#10*11*21] D h :.^e0'/c . D : s r /eCi3 .Dp.xe/S 
I- . (2) . (3) . D H :. 0*« C # . D„ . a* e £ : = . e s'* • 
1- . (1) . (4) . D h . Prop 

#40*2. H:k = A.D^'* = V 

h . #24*5*51 . D I- :. Hp . D : ~ (get) . a € k : 

[#10-53] D:(a):ae«.D.«ea: 

[#401] D'.xep'/c 

h. (1). #1011*21. DHHp. D .{x).xep' K . 
[#24*14] D.^«=V:Dh.Prop 

#40*21. h : * = A . D . *'* = A 
i)em. 

h. #24*51. Dh:Hp.D.~(a«).ae«. 

[#10*5.Transp] D . ~(ga) . a etc . xea . 



(2) 



(3) 

(4) 



(1) 



20—2 



308 MATHEMATICAL LOGIC [PART I 

[*40'll.Transp] D . x ~ e s'k (1) 

V . (1) . *1011'21 . D h : Hp . D . (x) . x ~ gs'k . 
[*24'15] D . s'k = A : D h . Prop 

In the above proposition, the two A's are of different types, since k is of 
the type next above that of s'k. Thus it would be more correct to write 
F:« = AnCls.D.s'«=AnV. 

But in the case of A it is not very important to keep the types distinct. 
#4022. h: A € K.D.p l K = A 

Y . #40-12 . D h : Hp . D . p'k C A . 
[#24-13] D.^'« = A:DI-.Prop 

In this proposition, the two A's are of the same type. 

*40221. H:Ve/e.D.s'*: = V 

Bern. 

\- . #40-13 . D h : Hp . D . V C s'k . 
[*24'141] D.s'* = V:DI-.Prop 

#4023. hgl/c.D.^Cs'K 

I- . #4012-13 .Dh:aeK.D.p'*Co.«C*'«. 
[*22'44] D.p'/cCs'/c: 

[#10-11-23] Dh:(aa).ae/c.D.^«Cs'/c:DI-.Prop 
Observe that the hypothesis g ! /c is essential to this proposition, since 
when k = A, p'k = V and s'k — A. Thus 

I- : g ! k. = . p'k C s'/e. 

#4024. f-:. a !/c:76/c.D y .y3C7:D./3C5^ 

K*4015. Dl-i^e/t.Dy./SCvrD./SC^A: (1) 

K #40-23. Dh-.RlK.D.p'/cCs'K (2) 

h . (1) . (2) . D h : Hp . D . /3 Cy* ,jp'« C s'k . 
[#22-44] D . /3 C s'k : D h . Prop 

The above proposition is used in the proof of #215-25. 
#40 25. H : x e s' k . = . g ! k « a (x e a) 

Dem. 

h . #22-33 .DF:g!Kn«(«e«). = . (37) .7e/c.7ea(a-ga). 

[#20*3] s . (37) .yeic.xey. 

[#40-11] = ■ 00 e s'k : D f- . Prop 

#40*26. h : g ! s'/c . = . (ga) . a e k . g ! a 



Dem. 



K #40-11 . D h:.g !«'*.= :(g#) :(ga).ae/c . #ea : 
[#11-23-55] = : (ga):a€* :(g#).a;ea: 

[#24-5] = : (ga) .ae/c.g!a:.Dh. Prop 



SECTION E] 



PRODUCTS AND SUMS OF CLASSES OP CLASSES 



309 



The following proposition is used in the proof of #216-51. 

$40*27. \-:.ar\s t K = A. = :y€K.D y .ar\y = A 

Bern. 

K #24-311.3 

I- :: a. r\ s'/c = A . = :. s'k C - a :. 

[#22'l - 35] =:.a.es'K.D x .cc~ea:. 

[#40*1] = :. (37) • 7 e k . x ey ."D x .x~ea.:. 

[#10*23] = :. ye k .xey. D x , y . #~ea:. 

[#11-2-62] = :. 7 e k . D y : x e 7 . D x . x ~ e a :. 

[#24'39] =:.7e/c.D 7 .an7 = A::Dh. Prop 

The following propositions are only significant when R is a relation whose 
domain consists of classes, for they concern p i R"a or s'i£"a, and therefore 
require that R"a should be a class of classes. 
#403. h.p'R"(a\j'/3)=p'R"af\p'R«0 [#37-22 . #40-18] 
#40-31. I- . s'R"(a u 0) = s'R"a u s'R"0 [#3722 .#40-171] 
#40-32. h .p'R"a yjp'R"p Cp'R"(a n £) 

Bern. 

Y . #37-21 . D K R"(a n £) C R"a n £"£ . 

[#40-16] D H .p'(i2"a n 22"£) C p'R"(a n £) (1) 

h . #40-17 . D r .p'R"a up'R"P Cp'(R«a n i2"/9) (2) 

h . (1) . (2) . #22-44 . D h . Prop 

#40-33. h . s'E"(a n/3)C s'R"a n «<£"£ [*37'21 . #40161 . #40-181] 

The following propositions no longer require that the domain of R should 

be composed of classes. 

#40-35. h . p l R ili K ~x-iPeic.Dfi.xe R"$) 

Bern. ^ 

h.*401 .Dh:.xep'R'"K.= :yeR'"fe. i 5 y .X€y: 

[#37-103] = : (3/8) . /3e * . 7 = £"/3 .D y .xey: 

[#10-23] =:$eic.y = R"p.'}f it y.xey: 

[#13191] =iPeK.? fi .xeR"P (1) 

h . (1) . #10-11 . #20-3 . D h . Prop 

#40-36. \-.s'R'"/c=$\(ft0).l3€K.xeR"/3} [Similar proof] 
#40-37. 1- . R"p'k Cp'R'"*; 

Dem. 

b . #37-1 . D I- :: x e R'tp'/e . = :. (g#) . y ep l K . xRy :. 

■ (33/) : e k . 3? . y e : xRy :. 
.(>&y):.(0) : e * .D .y e /3 : xRy :. 

• (0) ■> to) :0€K .D.yefS: xRy :. 

• (0) '- to) :0eK.D.yep. xRy :. 



[#40-1] 


= 


[#10-33] 


= 


[#11-26] 


D 


[#5-31] 


D 



310 



MATHEMATICAL LOGIC 



[PART I 



D :. (0) :. e k . D . (ay) . y e . xRy :. 

3:.(0):0eK.1.xeR"0z. 

D -..xep'R"'*:: D > . Prop 



[#10-37] 

[#37-1] 

[#4035] 

#40 38. h . i£' V« = s'R'"k 

Dem. 

V . *37'1 . D h :: a; e R"s'k . = :. (33/) .yes'ic. xRy :. 

[#40-11] = :. (ay) :. (ga) . a e /c . y e a : flj.By :. 

[*ll-6] = :.(aa):.ae/c:(ay).yea.a:%:. 

[*37l] =:.(aa).ae/c.a?ei2"a:. 

[#40-36] = :.xe s'R'"k ::Dh. Prop 

This proposition is frequently used in the proofs of arithmetical pro- 
positions. 

#40-4. \-:EllR«0.D.s'R"0 = x~{(Ry). ye0.xeR'y} 
This proposition is only significant when D'iJCCls. 

Dem. 

\-.*S1'Q.^V:B. V .O.R"0 = a{(^y).y € 0.a = R'y} (1) 

h.(l). #40-11. D 



(a*) : (32/) -ye0 •« = R ( y : a? e a : 

to) : y € & '• (a?) ■ «•= R'y -*««■: 

(ay) •ye/3.xe R'y : : D h . Prop 
H : E ! ! i2"£ . D . p'i2"£ = £ {y e £ . D„ . a? e 22'y } [Similar proof] 
H : (a?) . E'a; = P'x vQ'x.D .s'R"a = s'(P"a v, Q"a) = s'P"a w s'Q"a 



(1) 



h::Hp.D:.tfes<£"/3.= 
[#11-6] = 

[#14-205] = 

#40-41. 

#40-42. 

Dem. 

\- . #14-21 . D h : Hp . D . (x) . E ! R'x . E I P ( x . E ! Q'a; 

h.(l).*40-4.Dh :H.v.D.s'R"a = ${(ny).y€a.xeR'y} 

[Hp] = & '{(ay) .yea.xe P'yv Q'y] 

[#22*34] = x {(ay) lyeOLzxe P l y . v . x e Q'y} 

[*4*4.#10-42] — ^{(ay) .yeu.xeP'y . v .(^y) .yea.xeQ'y] 

[U).*40-4] = a jar e s'P"a . v . ar e s'Q"a] 

[*20-42.*22-34] =s'P"au*'Q"a 

[#40-171] = s'(P"a u Q" a ) : D h . Prop 

This proposition is used in #40*57, where we take R — C, P = T>, Q = CL 
#40-43. b :: E !! R"0 . D :. s'i2"/8 Co. = :y e £ . D v .R'y Cot 

Dem. 

h . #37-63 . 3 h :: Hp . D : . y e . D y . R'y C a :=: 7 e R"0 . D Y . 7 C a : 
[#40-151] = :s'i2"/3Ca::DKProp 

#40-44. I- :: E !! R"0 . D :. a C J9^"^ . = :y e .D y .aCR'y 
Dem. 

I- . #37-63 . D h :: Hp . D :. y e/3 . D y . a C R'y : = :-feR"0 . D y . a C 7 : ■ 
[#40-15] = : a C p'R"0 : : D K Prop 



SECTION E] PRODUCTS AND SUMS OP CLASSES OP CLASSES 311 

The following proposition is used in the proof of #84-44. 

*4045. Vi.ye$.?y.R t yCS t y:?.s t R"pCs t S< t P 

Dem. ■ ■ t x 

b . #1421 . D h :. Hp . O : E !! S"$ . E !! R"/3 : (1) 

[*37-62.*40'13] D-.yeP.Oy.S'yC s'S"/3 : 

[Hp] D:ye / 8.D J ,.i2'yO<S"/3: 

[#40-43.(1)] 3 : s'R"/3 C s'S"j3 :. D h . Prop 

The following proposition is used in the proof of #94*402. 
#40-451. h :. y e . D y . «'y C flf'y : 3 . p'£"£ Cp'flf"j8 
Dem. 
b . #14-21 . #37-62 . #40-12 . D b :. Hp . D : y e . D .^>'J2">S C E'y . 
[Hp] y.p'R"l3CS<y. 

[#40-44] 3 : j»*28"j8 C j»'iS"0 :. D h . Prop 

#40-5. \-.s'R"P = R"i3 

Dem. 

— * — > 

h . #3212 . #40-4 . D> . *'!*"£ = & {(ay) .yefi.xe R'y) 

[#32-18] = £ {(ay) - y e £ - *%} 

[(#37-01)] =i2"£.DKProp 

#40-51. H .^R"£ = £ {y e /3 . Dj, . #%} [#32-12 . #4041 . #32-18] 
— f 
p'R"& .is the class of terms each of which has the relation R to . every 

member of /3, just as R"fi is the class of terms each of which has the relation 
R to some member of /3. In the theory of series, p f R lt plays an important 
part, correlative to that played by R"fi (which is s'R"0, by #405). If is 
a class contained in a series whose generating relation is R, p'R"ft will be 
the predecessors of all members of j3, while R"j3 will be the predecessors of 
some /3. 

#40-52. b . s'£"/3 = R"fi [Proof as in #405] 

#40-53. > .p'R"^ = p{x€/3.D z . xRy\ [Proof as in #40-51] 
*4054. h . j/~K"£ = x (/3 C%x) [#4051 . #32-181] 

#40-55. b.p' 4 R"a = §(aCR<y) [*40'53 . #32-18] 

From this point onwards to *40'69, the propositions are inserted on 
account of their use in the theory of series. 

#40-56. h . 8 ( C"\ = F"\ [#33-5 . #405] 

In the above proposition, the conditions of significance require that X. 
should be a class of relations. 
#40 57. b . s'G"\ = s'(D"\ v d"\) = s'D"\ w s'CF'X [#4042 . #3316] 



312 MATHEMATICAL LOGIC [PART I 

#406. b . p'R"A = V . yS"A = V [*37'29 . *40'2 j 

#4061. b : a ! £ . D . p*R"$ C R"/3 . p'R"f3 C R"/3 
Dem. 

b. #37-73. D h : Hp.D. a !if"£. 

[#40-23] D . j9<~R"£ C s'l?"/3 . 

[*40-5] D . y£"/3 C R"/3 (1 ) 

Similarly H : Hp . D . p l R"$ C £"£ (2) 

h . (1) . (2) . D b . Prop 

#40*62. h : 3 ! yg . D . p'R«/3 C C'22 . p'R"/3 C C"i2 
[#40-61 . #37-1516 . #33161] 

The two following propositions (#40-63*64) are used in proving #40-65, 
which is used in #204'63. 

#4063. b-.Rip-a'R.D.pW'fi-A 
Dem. 

b. #3341. Transp. Db:x~ € a e R. D.R'x = A (1) 

b. #37-704. 2b:xe^. D.R'xell"/3 (2) 

b.(l). (2). *22'32. 2b iwe/3-a'R. 2. R'a:e'R"0.'R'a: = A. 

[*20'57] D.Aell"/3. 

[#40-22] D.p'£"/3 = A (3) 

I-. (3). #101 1-23. DKProp 

#40-64. h : a ! j3 - D'R . D . />'£"£ = A [Proof as in #4063] 

#40-65. h : a ! £ - C'22 . D .^^"^ = A .p'R"l3 = A [*40-6364 . #33-16] 

#40-66. I- :. a Cp<R"/3 . = :xea .ye /3 .D x , y . xRy 
Dem. 

b . #40-51 . D I- :: a Qp<R"l3 . = :. a C fc(y e/3 . D y . xRy) :. 

[*20'3] = :.x € a.2 x :ye^.D y .xRy:. 

[*ll-62] = :.(x,y):.xea.ye@.D. xRy :: D h . Prop 

<— — > 

#40-67. b:.@Cp'R"a.= : x € a .y e @ .2 x>y . xRy : = . uCp<R"/3 

[Proof as in #40*66] 

#40*68. I- . a rsp'P«a C P"j»'P"a 
Dem. 

b . #40-53 . Z> h :. x e a n p'P"a . D : x e a : y e a . D y . yPx : 
[*10-26] D : xPx :yea.D y . yPx : 



SECTION E] PRODUCTS AND SUMS OF CLASSES OP CLASSES 313 

[*10'24] D:(^z):zPx'.yea.D y .yPz: 

[*40-53.*37l05] D : x e P'yP"a :. D h . Prop 

This proposition is used in the theory of series (#206'2). 

#40-681. I- . a np'P"a C P"jt>'P"a [Proof as in #40-68] 
The following proposition is used in #211-56. 

#40682. h : a ! a n p<P"/3 . D . C P"a 

Dem. 

V . #4053 . D h :. Hp . D : (roc) '.xea:ye0.Z> y . yPx : 
[*5'31] O:(^x)iy€0.Dy.xea.yPx: 

[#11-61] O:ye0.^ y .{%x).xea.yPx. 

[#37-1] Dy.yeP^i.Dh.Prop 

#40 69. r : 3 ! C'P r» p' P"a . = . 3 ! P . 3 ! p'P"a 
Ztera. 

I- . #3324 . *24561 . D h : g ! C'P n p'P"a . D . 3 ! P . g ! #'P"a (1) 

K #40-62. Dh: 3 [!o.ayP"o.D.a!C f i J n/P"a (2) 

r- . #406 . D I- :. a = A . D : C'P « p'P"a = C'P : 

[#33-24] D:a!P.D. a !C'Pn i >'P" a (3) 

I- . (2) . (3) . #483 . Dh:a!P. a !p<P"a.D. a !C'Pnp<P"« (4) 
H.(l).(4). DKProp 

— > «— 

The above propositions concerning p'jR"£ and p'R"0 of course have 

analogues for *'J2"£ and *'22"£. But owing to #40-5, these analogues are 
more simply stated as properties of P"# and 22"£. Thus, for example, 
#37-264 is the analogue of *40'67. The above propositions concerning 

p'R"0 and p'R"0 will be used in the theory of series, but until we reach 
that stage they will seldom be referred to. 

#40-7. h . s'a % "0 = z {{rx, y).xecL.ye0.z = x%y) 

Dem. 

h. #40-11 .#38-3. D 

h.6-'a?"/3 = t{( a7 ,2/).ye/3.7 = ?2/" a .^e 7 } 
[#38-131] = z {(37, x,y).ye0.y = $ y"a .xea.z = x%y) 
[#13-19] =z\(<&x,y).xea.y€/3.z = x$y}.Db.Fr(yp 
This proposition is of considerable importance, since it gives a compact 
form for the class of all values of the function x%y obtained by taking x in 
the class a and y in the class 0. Thus, for example, suppose a is the class 
of numbers which are multiples of 3, and is the class of numbers which 
are multiples of 5, and xxy represents the arithmetical product of x and y, 



314 MATHEMATICAL LOGIC [PART 1 

then s'ax"^ will be the class of products of multiples of 3 and multiples 

of 5, i.e. the class of multiples of 15. Again suppose o and y8 are both classes 

of relations; then s'a|"/S will be all relative products R\8 obtained by 
it 

choosing R in the class a and S in the class /S. 
*40-71. V .s t %y"K = (s i K)%y = %y"s'K 

Dem. 

H . *40-38 . *38-31 . > I- . V ? y«K = ? y"s' K 

tt 

[*38- 2] = (s'k) ? y . D r- . Prop 

The hypothesis JB"aC«, which appears in *40-&-81, is one which plays 
an important part at a later stage. In the theory of induction (Part II, 
Section E) it characterizes a hereditary class, and in the theory of series it 
characterizes an upper section (when combined with a C C e R). 

*40-8. r :. a e k . O a . R"a C a : D . R"s'k C s'k 
Bern. 

h . *37'17l . D r- :: Hp . D :. a e k . 3 a : x e a . #lty . D X)!/ .yea:. 
[#11-62] ^:.aeK.xea.xRy .D a)Xt y.yea:. 

[*40-13] D a>Xjy .yes<K:. 

[#40'11 .*10'23] D :. x e s'k . xRy .D x>y .ye s'k :. 

[*37'l7l] 0:.R"s'KCs i K::D\-. : Prop 

«40'81. h :. a e k . D a . R"a C a : 3 . R"v'k C p'k 

Dem, 
h . *37*171 . Dh::.Hp. D::ae*. D ixea.xRy . D .^/ea:: 
[Exp.Comm] D ::#.% . D :. a eye . D : xea. D .y ea:. 

[*2'77] Oz.aeK.'y.xea-.Ozae/e.D.yea (1) 

h*.(l).*1011-21-27.D 

h ::. Hp . D :: #lfo/ . D :. a e /e . 3 a ■ # e a : D:ae/e.D a .;yea:. 
D z.xep'tc. D .yep'x :: 
[Imp] O : : a? ep'/e . a?ify .0 .ye p'k (2) 

h . (2) . *37-171 . D r . Prop 



#41? THE PRODUCT AND SUM OF A GLASS OF RELATIONS 

Summary, of *4il. 

The propositions to be given in this number, down to #41*3 exclusive, are 
the analogues of those of #40, excluding those from #403 onwards, which 
have no analogues. Proofs will not be given, in this number, when they are 
exactly analogous to those of propositions with the same decimal part in #40. 
The smaller importance of p'\ and s'X, as compared with p'X and s'X, is 
illustrated by the smaller number of propositions in #41 as compared with 
*40. 

Our definitions are 
#4101. p'X = $g(ReX.D R .xRy) Df 
#4102. s'X=^{(^R).ReX.xRy) Df 

Of the propositions preceding *41'3, which are analogues of propositions 
in #40, the only two that are frequently used are 
#4113. h-.ReX. y.RGs'X 
#41151. h : . s'X G S . = : R e-\ . Dr. RGS 

Of the remaining propositions of this number, which have no analogues 
in #40, the most important are #41'43-44*45, namely 

B's'X = s'D"\, d's'X = s'<I"X, C's'X = s'C'X 
These propositions are constantly required in the theory of selections (Part II, 
Section D) and in relation-arithmetic. Most of the other propositions of this 
number are used only once or not at all. 



#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 
#41 



01. p'X = $y(ReX.y R *xRy) Df 

02. s'\ = %§{(RR).Re\.xRy} Df 

I. b :.x(p ( X)y .= '.ReX.Dit.xRy 

II. h : a; (s'X) y. = . (gfi) .ReX. xRy 

12. \-:ReX.y.p'\GR 

13. t-zReX.D.RGs'X 

14. \-:ReX.x(p'X)y.D.xRy 
141. \-iReX.xRy.D.x(s'\)y 

15. \-:.SGp'X.= :ReX.D R .S(ZR 
151. h:.s<XGS. = :ReX.y R .RGS 

16. b-.XCfx.D.p'fiGp'X 
161. h zXCp.y.s'XGs'fjL 

17. h.p'Xv p'fi Gp'(Xn fi) 



316 



MATHEMATICAL LOGIC 



[PART I 



RQS:O s .xSy 



#41171. h . s'\ v s'/x = s'(\ u p) 

#4118. h .p'{\ u p) =p<\ f\p'ft 

#41 181. h . s'(X r\fj,)G s'\ n i'/i 

*4119. h::x(s { X)'t/. = :.R€\.Ojt 

*412. h:X.= A.D.p'X.= V 

*41'21. h : X = A . D . s'\ = A 

#4122. h:Ae\.D.^ = A 

#41-221. hV6\.D.a = V 

#4123. I- : a ! \ . D .£'\ G s'X 

h:.a!X:i2eX.D £ .i8fG22 : D.tfGs'X 
I- : x (s'\)y . = . 3 ! X. n 22 (o%) 
I- : a ! s<\ . = . (g R) .ReX.RlR 
\-:.P*s'\ = A. = :Re\.D B .PnR = A 
b.Gnv'p'^=P'Cnv"\ 



#41-24. 
*41-25. 
#41-26. 
#41-27. 
#413. 
Bern. 



K #31-13] . D 

I- :. y (Cnv'jo'X) x .= :x (p'X) y : 



[*41'1] 
[#31131] 
[*37-63.*3113] 
[*41-1] 

l-.Cnv'*'X. = *'Cnv"\ 
b.Cnv"p"/c=p"Cnv'"/c 
h . Cnv"s"/c = s"Cnx'"tc 
\-.s'a J \"\=a J \s'X 



= : R e \ . Dr . xRy : 

= :Re\.D B .y(Cnv'R)x: 

= :PeCnv"\.D P .yPx: 

= : y (p'Cnv"\) x :. D h . Prop 

[Proof as in #41-3] 
[#413 . #37-354] 
[#41-31 . #37-354] 



#41-31. 
#4132. 
#41-33. 
#41-34. 

Dem. 
V . #4111 . #38-13 . #13-195 .Dbt.x (s'a \ "A.) y . = 
[#35-1] = 

[#10-35] = 

[*4111.*35-1] = 

#41-341. h . s'f a"\ = (s'\) fa [Proof as in *41'34] 
#41-342. h.s'ta"A. = (s<A.)£a 
Dem. 

h . #361 1 . #35-21 .Dh.s't a"\ = s'a 1 " T «"* 
[#41-34] =a1(s<ra"X) 

[#41-341] =0^1(5^)^ 

[*36'11] = (s'\) I a . D h . Prop 



(nP).Pe\.x(a1P)y: 
(ftP) . P e X . x e a . xPy : 
xea: (a-P) • P e \ . #Py : 
x(a J ] s'\) y:.Dh. Prop 



SECTION E] THE PRODUCT AND SUM OF A CLASS OP RELATIONS 317 

The following proposition is used in #85*22. 
#4135. r . s'M I* "k = M \ s'k 
Dem. 

h . *4M1 . #3813 . D I- : x (s'Mf'/e) y. = . (get) . a e k . x (M [ a) y . 

[#35101] =.(Qa).a€/c.yea.xMy. 

[#4011.*35101] = . x (M\s'k) y : D h . Prop 

*41'351. \-.s"\M"k = {s'k) j \M [Proof as in *41'35] 

*414. h.D'_p*XC^'D"X 

Dem. 

V . #3313 . D 

V :: x eT>'p'X . = :. (gy) .x(p'X)y :. 
[#41-1] = :.(^y):ReX.D R .xRy:. 
[#11-61] D:.i2e\.D«.(ay).*%_:. 
[#3313] D:.R€\.D B .xeT>'R:. 
[*40'41. #331 2] D :. #ejt>'D"\ :: D h . Prop 

#4141. h.a'p<kCp f (I"\ [Proof as in #41-4] 

#41-42. h . C'p'X Cp'C'X 

Dem. 
h . #33132 . D h ::. x e C^<\ . = :: (gy) :x{p'\)y. v . y (£<\.)# :: 

[#411] = :: (gy) :: Re\ . D R .xRy : v : ReX. D s .yRx :: 

[#10-41-221] D::(^y)::(R)i.R€X.D.xRy.}f:Re\.D m yRx:: 

[#4-78] D :: (gy) ::(R)z. ReX.D : xRy . v . yRx :: 

[#11-61] D::(R)::R€\.D: (gy) : a-ity . v . yRx : 

[#33132] DzxeC'Rz: 

[*40-41.*33122] O :: x ep'C'X ::. D h . Prop 

#41-43. h . D 'a'X = .s'D"\ 

Dem. 

I- . #3313 .Dhz.xe T>'s'\ . = : (gy) . #(s'X)y : 

[#4111] =:(gy):(gi2). J K € X. a ; J Ky: 

[#1 1 -23-55] = : (gJ?) : 22 e \ : (gy) . arity : 

[#3313] =z(>giR).Re\.xeB<R: 

[*40-4.*33-12] = : x es'D"\ :. D r . Prop 

#41-44. K(T«'\ = *'<I"X [Proof as in #41-43] 
#41-45. r.(7<«'\. = s'C"\. 



De 



w. 



b . #3316 . D h . CVA. = D's'X u d's'X 
[#41-43-44] = s'D"\ u s<<3"\ 

[#40-57] = s'C'X . D H . Prop 



if 18 MATHEMATICAL LOGIC [PART I 

#41-5. ■\-.$P\\f t /**ip&\\ M p 

Dem. 

h.*34\L.3 

V x:x(p'\\p'fjL)z* = :.('&y) . x.(p'\)y.y(p'fi>)z :. 

[*41'1] =:.(<3.y):.P€\.Dp.<vPy:Q€p.D Q .yQz:. 

[#11-561 = :. (ay) :• (P> Q):PtX.D.xPy:Qep.O. y.Q* :. 

[♦11-87-39] D :. (ay) :• (P, : P«X. Qep.D.xPy *yQz :. 

[#11-61] D :. (P, Q) :.P« \ ..Q e ^ . D . (gy) . xPy .yQz. . 

[*34M] ,D.a:(P|Q).*:. 

[#13191] O :-.(P.Q, P) :.Pe\.Qep.R^P\Q.D.xRz :. 

[#11-21-35] D :. (P) : (gP, Q).Pe\. Q$p .R = P\Q. 3 . xRz : 

[#40*7] D :. (R) : Pes'X |'V- -> • ^^ : - 

[*4ri] O :. a; (#s'X J"/*).* i:..D H . Prop 

51(41 -51. K*'\|*V=:*VX|"/* 

H ■■#34-1 .3 

I- mx (s'\ | ■*'/*>* . = :. (gy). . a?(s'X) y . y (#/*)* :. 

[#41-11] s :. (gy) :- (3P) ~PeX . aPy : (aQ) .Qep.yQz :. 

[#11-54] = :..(ay) :• (HP, Q) = #«* ■ ^2/ .Qep.yQz :. 

[#ll-24-27] = :. (gP, Q) - (ay).P-«X .*Py - Qep.yQz:. 

[#1035] = :. (aP, Q) - P eX . Q ep .: ; (gy) . aPy, yQ* :. 

[#341] =:.(aP,Q):P€X.Q eAt ./r(P|Q)^:. 

[#13195] = :. (aP, Q,R) .Pe\.Qep.R = P \Q.xRz :. 

[#U-24.*40-7] = :. (gP) • #« **X j'V* • ^ : - 
< [#41*11] = :.x (s's'X | "a*) * " 3 •"' -Prop 

The above proposition, which is used in #92-31, states that, if X and /a are 
classes of relations/the relative product of the relational sum of X and the 
relational sum of p is the relational sum of all the relative products formed 
of a member of X and a member of p. 

The following proposition is used in #96-111. 
#41-52. h:.o1*'VCQ.= :.Pe\.Dp..dTP'CQ 



Dem. 



K*35'l..*4i'11.3 

l- sra.|^X G Q . =u x ..*« : (aP) • P e X . aPy : D^ . a% :. 
[#10-35-23] = :. x e a . P e X . aPy . Op,*,, . a% :. 
[#35-1] = :. P e\.x(a^P)y .^ £fmu ..xQy:. 

£#11-62] = :. P e X . Dp . a] P G Q :: D h . Prop 



SECTION E] THE PRODUCT AND SUM OP A CLASS OF RELATIONS 

The following proposition is used in *162'32 and in *1 66461. 
*41-6. H : . y e £ . V P'y = Q'# o B'yi O - *'P"£ = W£ c PR"? 

Dem. 
K *376 . *1421 . *41-11 . *13195 . 3 



319 



h::Hp.D:.w(s f P" i 8)v.= 

[Hp] 

[*23-34.*10-42] •= 

[*3r-6.*4111] = 



(&y)'ye0~u(Q'yK>R'y)v: 

(ay) ■ y € £ • «-(Q'y>» • v ■ (ay) - y e £ . « (£'y) »-. 

w (s'Q"A)» ■ v • M (*'-R"j8) v : : D+ . Prop 



#42. MISCELLANEOUS PROPOSITIONS 

Summary q/"#42. 

The present number contains various propositions concerning products and 
sums of classes. They are concerned chiefly with classes of classes of classes, 
or with relations of relations of relations. These are required respectively in 
cardinal and in ordinal arithmetic. Thus #421 is used in #112 and #113, 
which are concerned with cardinal addition and multiplication, while #4212"2 
are used in #160 and #162, which are concerned with ordinal addition. #42*22, 
though not explicitly referred to, is useful in facilitating the comprehension of 
propositions on series of series of series, or rather on relations between relations 
between relations, which are required in connection with the associative law 
of multiplication in relation-arithmetic. 



#421. H . s's"k = s's'k 

Here k must, for significance, be a class of classes of classes. The proposi- 
tion states that if we take each member, a, of k, and form s'a, and then form 
the sum of all the classes so obtained, the result is the same as if we form the 
sum of the sum of k. This is the associative law for s, and is (as will appear 
later) the source of the associative law of addition in cardinal arithmetic. The 
way in which this proposition comes to be the associative law for s may be 
seen as follows: Suppose k consists of two classes, a and /3; suppose a in turn 
consists of the two classes £ and rj, and fi of the two classes £' and ij . Then 
s <a = | u v ■ *'/3 = I' w y> (This will be proved later.) Thus s"k has two 
members, one of which is ^uij, while the other is f ' «-> rj '. Thus 

s's"k = (^w)j)u (£' u v '). 
But s'k has four members, namely £, 77, £', rf. Thus s's'k = f w 17 u f u 1/ . 
Thus our proposition leads to 

(£ « V) w (£' w V)= £ w V w f w ri, 
which is obviously a case of the associative law. 

Our proposition states the associative law generally, including the case 
where the number of brackets, or of summands in any bracket, is infinite. 
The proof is as follows. 

Dem. 

b . #404 . D h : : x e s's"tc . = 



[#4011] 
[#11-6] 
[#4011] 
[#4011] 



. (get) . ae/c .xes'a :. 
■(aa):ae«:(a£).£ea.ae£:. 

• (a£) =• (3«) ■ a e K • f e a : x € f '•■ 
.(a£).fe*'iie.a-ef :. 
. x e s's'k :: D r- . Prop 



SECTION E] MISCELLANEOUS PROPOSITIONS 321 

#4211. V,.'p i p i ' t K—p t s t K 
Dem. 

b . #40'41 . D I- :. x ep'p"ic . = : P e k , Dp . x ep*p : 

[*40-l.*ll-62] = : P € k . y e p . D^ .#67: 

[*ll-2.*10-23] = : (ftp) .pete. yep. Dy.xey: 

[#4011] =:yes { K .Dy.xey: 

[#401] = : x ep's'ic :. D h . Prop 

This is the associative law for products. Supposing again, for illustration, 
that k consists of the two classes a, P, while a consists of the two classes £, rj 
and P of the two classes f , rj', then p"ic consists of the two classes £ r\ rj and 
f ' r\ rj, so that p l p li K = (£ c\ rj) n (f r» 17'), while pV/e = f r\ rj r\% r\ rj. Thus 
our proposition becomes 

(%r\rj)r*(? r\rj')=%r\rjc\g r\rj. 

A descriptive function R'k whose arguments are classes or classes of classes 
may be said to obey the associative law provided 

R'R"k = R's'k. 

This equation may be interpreted as follows: Given a class a, divide it 
into any number of subordinate classes, so that no member is left out, though 
one member may belong to two or more classes. Let the classes into which 
a is divided make up the class k, so that k is a class of classes,- and s* k = a. 
Then the above equation asserts that if we first form the R's of the various 
sub-classes of a, and then the R of the resulting class, the result is the same 
as if we formed the R of a directly. 

In some cases — for example, that of arithmetical addition of cardinals — 
the above equation holds only when no two members of k have a common 
term, i.e. when the parts into which a is divided are mutually exclusive. 

For a descriptive function whose arguments are relations of relations, we 
shall find another form for the associative law; this form plays in ordinal 
arithmetic a part analogous to that played by the above form in cardinal 
arithmetic. 

#4212. Ks's"X. = sV\ 

Devn. 

K*41~ll .DF:«(«'«"X)j/.= . (g>) . fi e\ . x (s* fi) y . 

[*41'11] =.(ftfi,P). l ie\.Pe f i.xPy. 

[#40-11] = . (aP) .Pes'X. xPy . 

[#41-11] = . x (sV\) y : D 1- . Prop 

#4213. H .p'p"\ = p <s'\ 

Dem. 

K*41'l .Dh :.x(p'p"\)y . = :/j.e\. D IIL .x(p t fi)y. 

[#41*1] =: fieX.Re/x.D^s.xRy: 

b&w i 21 



322 MATHEMATICAL LOGIC [PART I 

[*ll-2.*10-23] =:(Rfi).tJL€\.Refi.D B .a;Ry: 

[*40'11] = : R e s'X . D^ . xRy : 

[*41 1] = : x (p's'X) y : . D H . Prop 

*42 2. I- . C's'C'P = s'C"C'P = F"CP = F*'P 

This proposition assumes that P is a relation between relations. For 
example, .suppose we have a series of series, whose generating relations are 
ordered by the relation P. Then CP is the class of these generating relations ; 
s'C'P is the relation "one or other of the generating relations which compose 
C'P" and C's'C'P is the class of all the terms occurring in any of the series. 
C'C'P is the fields of the various series, and s'C'C'P is again all the terms 
occurring in any of the series. F"C'P is all the terms belonging to fields of 

series which are members of C'P, and F 2 'P is all members of fields of members 
of the field of P; each of these again is all the terms occurring in any of the 
series. The proof is as follows: 

Dem. 

V . *41'45 . D h . C's'C'P = s'C'C'P (1) 

h . *40-56 . D h . s'C'C'P = F"C'P (2) 

K*33-5. Dh.F"C'P =F <7 F'P 

[*37'38] =~F*<P (3) 

K(l).(2).(3).DKProp 

The following propositions apply to a relation of relations of relations. 
These propositions are useful for proving associative laws in ordinal arith- 
metic, since these laws deal with series of series of series, and series of series 
of series are most simply constituted by supposing the generating relations of 
the constituent series to be ordered by relations which are themselves ordered 
by a relation P. 

*42-21. K s'C'"C"CP = C" s'C'C'P = C'Cs'CP = C'F'.'C'P = C"~F* 'P 

Dem. 

h . *4038 . D H . s'C'C'CP = C's'C'CP (1) 

K (1) . *42-2 . D r . Prop 

*42 22. h . s's'C'C'C'P = s'C's'C'C'P = s'C'C's'CP 

= C's'C's'C'P = s'C"F"CP 

— » — > 

_ jpajptiQip _ ]?«]?* tp = jps tp 

[*42-21 . *41-45 . *40-56 . *42'2 . *37'3] 

If P, in the above proposition, is a relation which generates a series of 
series of series, the above gives various forms for the class of ultimate terms 
of these series. Thus suppose Q eC'P; then Q is a relation between generating 



SECTION E] MISCELLANEOUS PROPOSITIONS 323 

relations of series. If now R eC'Q, R is the generating relation of a series 
which we may regard as composed of individuals. The class of individuals so 
obtainable may be expressed in any of the above forms, as well as in others 
which are not given above. 

*42 3. V . sV'i2"a = s'R"ol 
Dem. 

h . *421 . D h . s's"R"a = s's'R"a 

[*40-5] =s'E"a.Dh.Prop 

*42-31. h . s's"R"a = s*R"a [Proof as in *42'3] 



21—2 



#43. THE RELATIONS OF A RELATIVE PRODUCT 
TO ITS FACTORS 

Summary q/'#43. 

The purpose of the present number is to give certain propositions on the 
relation which holds between P and Q whenever P — Q | R, or whenever 
P = R\Q, or whenever P = R | Q \ S, where R and S are fixed. In virtue of 
the general definitions of #38, these relations are respectively j R, R\, and 
(R |) | ( j S). Such relations are of great utility both in cardinal and in ordinal 
arithmetic; they are also much used in the theory of induction (Part II, 
Section E). In place of the notation (R \ ) | ( | S), which is cumbrous, we adopt 
the more compact notation R\\S. If X is a class of relations, R | "A, will be the 
class of relations R \ P where P e \ | R"\ will be the class of relations P \ R 
where Pe\, and (R || S)"\ will be the class of relations R | P \ S where Pe\. 
These classes of relations are often required in subsequent work. 

In virtue of our definitions, we have 
*43112. \-.(R\\S)'Q = R\Q\S 

The propositions most used in the present number (except such as merely 
embody definitions) are the following: 

*43'302. h.(P).Pe<I<(R\\S) 

*43-411. K.R'"a"\ = a"|JR"\ 
*43'421. h.s'\ R"\ = (s<\) | R 

The remaining propositions are used seldom, but their uses, when they are 
used, are important. 

*4301. R\\S = (R\)\(\S) Df 

At a later stage (in #150) we shall introduce a simpler notation for the 

special case of R\\R. The following propositions are for the most part. 

immediate consequences of the definitions, and proofs are therefore usually 

omitted. 

#431. \-:P(R\)Q. = .P = R\Q 

#43101. \-:P(\R)Q. = .P = Q\R 

#43102. \-:P(R\\S)Q. = .P = R\Q\S 

#4311. \-.R\'Q = R\Q 
#43111. h . | JK'Q = Q | JB 
#43112. h.(R\\SyQ = R\Q\S 
#4312. h.E!E|'Q 



SECTION E] RELATIONS OF A RELATIVE PRODUCT TO ITS FACTORS 



325 



#43121. KEIJJR'Q 
#43122. b.E I (R\\S)'Q 
*43'2. b.(R\)\(S\) = (R\8)\ 

Bern. 

h.*4 ! ^l.D\-:L{( < R\)\(8\)}N. = .(^M).L = R\M.M = S\N, 

[*13195.*34-21] = .L = R\S\N. 

[#43-1] ^.L{(R\S)\}N:Db.¥vo V 

#43-201. I- . (| R) | (| S) = | (8 1 R) [Proof as in #43-2] 

#43-202. \-.(\R)\(8\) = (S\)\(\R) = 8\\R [Proof as in *43"2] 

#43-21. b.(P\\Q)\(R\) = (P\R)\\Q 

#43-211. b.(R\)\(P\\Q) = (R\P)\\Q 

#43-212. b.(P\\Q)\(\R) = P\\(R\Q)- 

#43-213. \-.(\R)\(P\\Q) = P\\(Q\R) 

#43-22. H.(P||Q)|(i2||fl)-(P|B)||(^|Q) 

#43-3. K(P).Pe<Pi2| [#4312. #33-43] 

#43301. h.(P).P€a < | J R ( 

#43-302. \-.(P).Pe<I'(R\\S) 

#43-31. KPf(RR| = PrC'i2| = P 

Dem. 

h . #4312 . #33-431 . D I- . d'P C a f ,K | 

[#33-161] DK(I<PCC<i*| 

I- . (1) . (2) . #35-452 . D h . Prop 

#43311. H.Pr(F|£ = PrC'|£ = P 
#43-312. b.P[a ( (R\\S) = P[C t (R\\8) = P 



(1) 
(2) 



#43-34. [-.^1^ = 1^^ = ^ 
#434. h.iJ"D'P = D'i2|'P 

#43-401. H . R"(I<P = a 4 1 J?'P 
#43-41. h . 22<"D"\ = T>«R \ "\ 

#43-411. h . iz'"<I"\ = a" | R"\ 
#43-42. Ks<.R|"\ = 22|s f X 
Dem. 



[*4311111] 
[#37-32. #43-1] 

[#37-32 . #43101] 
[#43-4 . #37-355] 

[#43-401 . #37-355] 



I- . #4111 . #371 . #431 . D 



h:.a;(s'E|"\)2.= 
[#34-1] = 

[#11-6] = 

[#41-11. #341] = 



{^T).Te\.x{R\T)z: 
{^T):Te\:(^y).wRy.yTz: 
(^-.xRyz^Ty.TeX.yTz: 
^ORIi'X^r.DKProp 



326 MATHEMATICAL LOGIC [PART I 

#43421. r- . s< j R«\ = (i*\) j R [Proof as in #43-42] 
#4343. h.s<(R\\S)"\ = (R\\Sys<\ 
Dem. 

H . #37-33 . D K «<(£ || £)"\ = s'R | " | S"X 

[*43'42] =5|(8'ifif"X) 

[#43-421] = i2j£'A,|,Sf 

[#43112] =(R\\Sys t X. Dh.Frop 

#4348. l-:D f PC a .D.Q|'P=.(Q^ a )|'P [*35'481] 
#43-481. H : (I'P C . D . 1 22'P = 1 08 1 P)'P [*35'48] 
#43-49. l-:s'D"\Ca.D.(Q|)[ k \={(Q| k a)|}| k \ 
.Dew. 

K #40-43. Dhr.Hp.DrPeX.D.D'PCa. 

[*43-48] 3-Q|'-P={(Qr«)|}'P (1) 

h . (1) . #35-71 . D h . Prop 

#43-491. \-:s<a"\C/3.D.QR)t\={\(l3'\R)}r\ [Proof as in #43-49] 
#43-5. l-:iyPCa.a<PC0.3.(Q\\RyP={(Q\- a )\\(i31R)yp 
[#35-48-481. #43-1 12] 

#43-51. t-:s<D«\Ca.s<a"\C/3.D.(Q\\R)r\ = {(Qra)\\(/3'\R)}\>\ 
Dem. 
h . #40-43 .DI-:.Hp.D:PeX.D. D'P C a . d'P C £ . 

i* 43-5 ] 3-(Qii^)^={(Qr«)iK/8ii2)}'p a) 

f- . (1) . #35-71 . D h . Prop 

The above proposition is used in the proof of #74-773. 



PART II 

PROLEGOMENA TO CARDINAL ARITHMETIC 



SUMMARY OF PART II 

The objects to be studied in this Part are not sharply distinguished from 
those studied in Part I. The difference is one of degree, the objects in this 
Part being of somewhat less general importance than those of Part I, and 
being studied more on account of their bearing on cardinal arithmetic than 
on their own account. Although cardinal arithmetic is the goal which 
determines our course in Part II, all the objects studied will be found to be 
also required in ordinal arithmetic and the theory of series. As this Part 
advances, the approach to cardinal arithmetic becomes gradually more marked, 
until at last nothing is lacking except the definition of cardinal numbers, with 
which Part III opens. 

Section A of this Part deals with unit classes and couples. A unit class 
is the class of terms identical with a given term, i.e. the class whose only 
member is the given term. (As explained in the Introduction, Chapter III, 
pp. 76 to 79, the class whose only member is x is not identical with x.) We 
define 1 as the class of all unit classes, leaving it to Part III to show that 1, 
so defined, is a cardinal number. In like manner, we define a (cardinal or 
ordinal) couple, and then define 2 as the class of all couples. The propositions 
on couples will not be much referred to in the remainder of the present Part, 
since their use belongs chiefly to arithmetic (Parts III and IV). On the other 
hand, the properties of unit classes are constantly required in Sections C, D, E 
of this Part. 

Section B deals, first, with the class of sub-classes of a given class, i.e. of 
classes contained in a given class. The sub-classes of a given class are often 
important in arithmetic. Next we consider the class of sub-relations of a 
given relation, i.e. relations contained in a given relation. The propositions 
on this subject are analogous to those on sub-classes, but less important. 
Next we consider the question of "relative types," i.e. taking any object x, and 
calling its type t'x, we give a notation for expressing in terms of t'x the type 
of classes of which a; is a member, or of relations in which x may be either 
referent or relatum, and so on. The notations introduced in this connection 
are very useful in arithmetic, especially in connection with existence-theorems. 
But the propositions of Section B are very seldom required in the later sections 
of the present Part. 

Section C, which deals with one-many, many -one and one-one relations, 
is very important, and is constantly relevant in the sequel. A relation is 
one-many when no term has more than one referent, many-one if no term has 
more than one relatum, and one-one if it is both one-many and many-one. 



330 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

In this section, we define the notion of similarity, upon which all cardinal 
arithmetic is based: two classes are said to be similar when there is a one-one 
relation whose domain is the one and whose converse domain is the other. 
We prove the elementary properties of similarity, including the Schroder- 
Bernstein theorem, namely: If a is similar to part of j3, and yS is similar to 
part of a, then a is similar to fB. 

Section D deals with the notion of selections, upon which both cardinal 
and ordinal multiplication are based. A selection from a set of classes is 
a class consisting of one member from each class of the set. Thus a selective 
relation R may be defined as one which, for a given class of classes k, makes 
R'a. a member of o whenever a is a member of tc. More exactly, a selective 
relation for a class of classes k is one which is one-many, which has k for its 
converse domain, and is such that, if ocMa, then x e a. Such a relation may 
be called an e-selector from k. More generally, we may define a P-selector 
from «asa relation which is one-many, which has k for its converse domain, 
and which is contained in P. The theory of selectors is very important in 
arithmetic. But until we come to cardinal multiplication in Part III, Section B, 
the propositions of this fourth section will seldom be relevant. 

Section E deals with mathematical induction, not in the special form in 
which it applies to finite integers (this is considered in Part III, Section G), 
but in a general form in which it applies to air relations. The propositions 
of this section are of very great importance, primarily in the theory of finite 
and infinite (Part III, Section C, and Part V, Section E), but also in many 
other subjects, and especially in the derivation of series from one-many, 
many-one or one-one relations — for example, in ordering the "rational" points 
of a projective space by means of successive constructions of harmonic points. 
The ideas involved in this section are somewhat complicated, and we must 
refer the reader to the section itself for an account of them. 



SECTION A 

UNIT CLASSES AND COUPLES 

Summary of Section A. 

In this section we begin (#50) by introducing a notation for the relation 
of identity, as opposed to the function "x = y"; that is, calling the relation of 
identity /, we put 

I == $§( x = y) Df. 

The purpose of this definition is chiefly convenience of notation. The 

definition enables us to speak of/, D'Z", I\B,a^I, I" a, etc., which we could 
not otherwise do. 

At the same time we introduce diversity, which is defined as the negation 
of identity, and denoted by the letter J. The properties of / and J result 
immediately from #13, since 

xly . = . x = y. 

We next introduce a very important notation, due to Peano, for the class 
whose only member is x. If we took a strictly and purely extensional view of 
classes, we should naturally suppose this class to be identical with x. But in 
view of the theory of classes explained in #20, it is plain that x can never be 
identical with a class of which it is a member, even when it is the only member 
of that class. Peano uses the notation "ix" for the class whose only member 
is x; we shall alter this to "i'x," following our general notation for descriptive 
functions. Thus we are to have 

l'x = $(y = x) = $ (ylx) = I'x. 
Hence we take as our definition 

t=7 Df, 

since this definition gives the desired value of i'x. The properties of i are 
many and important. 

It is important to observe that "i'a" means "the only member of a." Thus 
it exists when, and only when, a has one member and no more, in which case 

a is of the form i'x, if x is its only member. Thus "I'a" means the same as 
»»» 

"(ix)(x €<*)," and "i l z{$z)" means the same as " (ix) (<bx\" What we call 
"i'a" is denoted, in Peano's notation, by "7a." 

Classes of the form i'x are called unit classes, and the class of all such 
classes is called 1. This is the cardinal number 1, according to the definition 
of cardinal numbers which will be given in #100. The properties of 1, so far 
as they do not depend upon other cardinals, or upon the fact that 1 is a 
cardinal, will be studied in #52. 



332 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

After -a number (#53) containing various propositions involving 1 or i, we 
pass to the consideration of cardinal couples (#54) and ordinal couples (#55). 
A cardinal couple is a class i l x u i'y, where x =J= y. The class of such couples 
is defined as 2, and will be shown at a later stage (#101) to be a cardinal 
number. An ordinal couple, which, unlike a cardinal couple, involves an order 
as between its members, is defined as a relation i'x j" i'y (cf. #35*04), where 
we may either add x =f y or not. The properties of ordinal couples are in part 
analogous to those of unit classes, in part to those of cardinal couples. In #56, 
we define the ordinal number 2 (which we denote by 2 r , to distinguish it from 
the cardinal 2) as the class of all ordinal couples t'x f i'y, where x ={= V- It will 
be shown at a later stage that this is an ordinal number according to our 
definition of ordinal numbers (#153 and #251). 



*50. IDENTITY AND DIVERSITY AS RELATIONS 

Summary o/*50. 

The purpose of the present number is primarily notational. For notational 

reasons, we must be able to express identity and diversity as relations, and not 

merely as propositional functions, i.e. we require a notation for &$ (x = y) and 

£§(x^y). We therefore put 

/ = ^(tf =2/ ) Df, 

J=^I Df. 

In spite of the fact that diversity is merely the negation of identity, the 
kinds of propositions that employ diversity are quite different from the kinds 
that employ identity. Identity as a relation is required, to begin with, in the 
theory of unit classes, which is our reason for treating of it at this stage. It 
is next required, constantly, in the theory of mathematical induction (Part II, 
Section E). It is required also in showing that cardinal and ordinal similarity 
are reflexive. These are its principal uses. 

Diversity, on the other hand, is required almost exclusively in the theory 
of series (Part V), and the first number in that theory will be devoted to 
diversity. Until that stage, diversity will seldom be referred to, with one 
important exception, namely in proving the associative law of multiplication 
in relation-arithmetic (#174). 

The most important propositions on identity in the present number are the 
following: 

*5016. h . I"a = a 

*504. b.R\I = I\R = R 

*50'5. \-.a J \I = I[a = a J \I[a 

*50-51. KCnv'telJ^a-J/ 

*5052. \-.T><(a'\I) = a<(a'\I) = C'(a'\I) = GL 

*50*62. hid'RCa. 2. R\(I\-a) = R 

*5063. hD'EC a. D.Ila\R = R 

The most important propositions on diversity in the present number are 
the following: 

*5023. \-:RGJ. = .RGJ 

*50-24. h:EG/. = .(a;).~ (xRx) 

*50-43. \-:R*aj. = .RfsR = k 

*50'45. h-.R^dJ.D.RdJ 

*S0*47. h :. iJ 2 G R . D : i£ G J . = . R? G J . = .RnR=A 



334 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

It will be observed that all these propositions are concerned with R G J or 
R*Q,J, both of which are satisfied if 22 is a serial relation. The hypothesis 

R*(ZJ or Rr\R = A characterizes an asymmetrical relation, i.e. one which, if 
it holds between x and y, cannot hold between y and x. 



*5001. I = x§(x = y) Df 

*5002. J=-^I Df 

Most of the propositions of this number are obvious, and call for no 
comment. 

*501. Y:xly.= .x = y [*213 .(*50-01)] 

*5011. Y-.xJy.-.x^y [*23'35 . *501 . (*5002)] 

*5012. h. J=w§(x$y) [*5011 . *21'33] 

*50 13. h . a ! I [*13-19 . *10'24-281 . *50-l] 

*5014. \-.I'y = y [*30-3 . *50 1 . *1011] 

*5015. r . (y) . E ! I'y [*50'14 . *1421 . *1011] 

*5016. KJ"a=a 

Bern. 

V . *37l .D\- :xe J"a . = . (g#) .yea. xly . 
[*50-l] = .(^y).yea.x = y. 

[*13195] =.« € a:Df-.Prop 

*5017. \-:.xea.O x .R'x = x:D.R"a = a 

Bern. 

h . *1421 . D V : Hp . Z> . E !! R"a (1) 

h . *5014 . D h :. Hp . D : xea . D x . R'x^I'x : 

[*37'69.(1)] D :J B"a = /"a: 

[*5016] D : R"a = a :. D h . Prop 

*502. h.I=I 
Bern. 

r . #50*1 . D r : xly . = ,x = y . 

[*13'16] =.y = x. 

[*50-l] =.ylx. 

[*31'11] = . xly : D h . Prop 

*5021. h.J=J 

I- . *21-2 . (*50-02) . D h . «/= ^- 7 (1) 

[*50'2.*23-83] =-i-7 

[*31 : 16] =Cnv'-r-7 

[(l).*31-32] = ,7.3 K Prop 



SECTION A] IDENTITY AND DIVERSITY AS RELATIONS 335 

#5022. \-:RQI. = .RQI [*314 . *502] 

#5023. \-:RGJ. = .RGJ [*314 . *50 : 21] 
*5024. hi?GJ.H.(*).~ (xRx) 

Dem. 

V . #5011 . D I- : . R G / . = : xRy .D Xty .x^y: 

[Transp] = : x = y . D x> y . ~ (xRy) : 

[#13191] = : (a?) . ~ (xRx) :. D h . Prop 

#503. V.{x).xlx [*50\L . *1315] 

#50 31. h . B f I = V . d'l = V 

Bern. 

h . #50-3 . *1024 .Dh:.(«): (gy) . xly :. (a?) : (gy) . y/a; :. 

[#3313131] Dh(a;).iceD'/:(^).«6a'/: 

[#24-14] Oh.D'/=V.(F/=V.DKProp 

*5032. h . CI = V [*50-31 . *3316 . #24-27] 

#50-33. H : g ! J. D . D'J= V . (1'/= V . CV= V 
Dew. 

h.*13171 .Transp.Dbz.y^z.D-.x^y.v.x^z:. 
[*50'11] Dbz.yJz .DzxJy.v ,xJz\ 

[*33'14] DzxeD'J (1) 

K (1) . *llll-35 . D I- : g ! J. D . « e D'/: 
[#1011-21] DhglJ.D.^.^eD'/. 

'[#24-14] D.D'«/=V (2) 

h.(2).*50-21 ; DI-.Prop 

In the above proposition (#50-33), the hypothesis g ! J is equivalent to 
the hypothesis that more than one object exists of the type in question. This 
can be proved for all except the lowest type. For the lowest type, we can 
only prove the existence of at least one object: thi3 is proved in #24-52. For 
the next type, we can prove the existence of at least two objects, namely A 
and V; these are distinct, by #24-1. For the next type, we can prove the 
existence of 2 2 objects ; for the next, 2 4 , etc. But for the class of individuals 
we cannot prove, from our primitive propositions, that there is more than 
one object in the universe, and therefore we cannot prove g ! J. We might, 
of course, have included among our primitive propositions the assumption 
that more than one individual exists, or some assumption from which this 
would follow, such as 

(3<£> x,y).<l>\x.~$\y. 
But very few of the propositions which we might wish to prove depend upon 
this assumption, and we have therefore excluded it. It should be observed 
that many philosophers, being monists, deny this assumption. 



336 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#5034. h.g!/£Cls 

Dem. 

h . #2041 . #2238 . (#24-01 02) . D h . A, V e Cls . 

[*24'1] DKA + V.A, VeCls. 

[*36-13.*5011] 3 H ■ A { J £ Cls] V . 

[#1024] 3 h ■ Prop 

#50 35. h . g ! / 1 Rel [Proof as in *50'34] 

#504. \-.R\I = I\R = R 

Dem. 

\- . #341 . D b : x(R \ I)z . = . (gt/) . xRy . ylz . 

[#50-1] = 

[*13'195] = 

K*34-1.DH:#(J|P)*.= 

[*50-l] s 

[#13195] s 

K (1) . (2) . D H . Prop . 

*50'41. h:P|PGJ.5.P|PG,7. = .P*P = A 

Pew. 

h . #341 . #5011 .Dh:.P|PGJ.= : (gy) ■ ®Ry • V?* .^ x , z .x^z: 
[#13-196] = :(x): ~ (gy) . xRy .yPx : 

[#10'252] =:~(>&x,y).xRy.yPx: 

[*31-11] = :~(<&x,y).xRy.xPy: 

[*23-33.*2551] = :2*AP = A: (1) 

[#31*14-24] ==:jR*HP = A: 

p,p 



(>&y).xRy.y = z. 
xRz 

(32/) • aTy ' y Rz m 
(fty).x = y.yRz. 

xRz 



(1) 



(2) 



(1) 



R,P 



= iR\Cnv<PGJ: 



= iR\PdJ 



[*34-203] 

V . (1) . (2) . D h . Prop 

#50-42. KZ 2 = I 

Pew. 

I- . #34-5 . D h : #J 2 .z . = . (gy) . a?iy . ylz . 

[*5o-i] =.(w)' xI y-y= z - 

[#13-195] s.s/stDKProp 



(2) 



#50*43. h:22 2 GJ'.= ..K«jR = A 



f*50-41 11 



This proposition is useful in the theory of series. "Rr\R = k" is the 
characteristic of an asymmetrical relation. 



SECTION A] IDENTITY AND DIVERSITY AS RELATIONS 337 

#5044. h:a!(JKA/).D.a!(i?n/) 
Dem. 

h . #2333 . #501 . D H : 3 ! (R r» I) . = . fax, y) . xRy .x = y. 

[#13195] = . fax) . xRx . 

[#34-54] D.fax).xR*x. 

[*13'195] D . fax, y) . xR?y . x = y . 

[*2333.*50-l] D . a ! (J? 2 *I) : D h . Prop 

#5045. !- : i2 2 G «/ . D . R G J [*5044 . Transp . *25'311] 

#5046. (-:i2n J R = A.D.i2G/ [#50-43-45] 

#5047. h:. J R 2 G J R.D: J KGJ. = ..R 2 GJ r . = .iJn J R = A 

r . #2344 . D h : . Hp . D : R G / . D . E 2 G J (1) 

I- . (1) . *50-45-43 . D H . Prop 

This proposition is used in the theory of series. If R is a serial relation, 
we shall have JR 2 G R and RQ.J. 

#505. h.a1/=/ra = a1/r« 

Dem. 

V .#351 .Oh :x(a J ]I)y . = . xea.xly . 

[#50 - l] = ,xea.x = y . 

[#13193] =.yea.x = y. 

[*50"1] = .xly.yea. 

[#35-101] =.x(I[a)y (1) 

K(l).*23-5.DI-.a1/ ^In/fa 

[*35'11] =a > |/[ k a (2) 

f-.(l).(2).Dh.Prop 

#50-51. h.Cnv'(a > I/) = a1J [#35-51 .#50-2-5] 

#50-52. h . D'(a 1 /) = <3'(a 1 /) = C"(a 1 /) = a 
i)em. 

I- . #35-61 .Dr. D'Cal I) = a a D'l 

[#50-31] = a n V 

[#24-26] =a (1) 

Similarly \- .a t (a J \I) = a (2) 

r . (1) . (2) . #33-18 .Dr. Prop 

#50-53. h.a > |/^ = (an/3)T/ = /p(an / g) 

I- . #35-21 . #50-5 .D\-.a J \I\-/3 = a J \(/3 J \I) 
[#35-32] =(an j 8) > |/ (1) 

r . (1) . #50-5 .Dr. Prop 
R&w I 22 



338 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#5054. h.(«1/) 2 = a1/ 

Dem. 

h.*50-5.DK(a1/) 2 ==(a1/)|(ir«) 
[#35-12] =o1/ 2 r« 

[#50-42] =aVr« 

[#50-5] =a > |/.p h.Prop 

#5055. h:an/3=A. = .at^GJ 

Dem. 

V . #2437 . #5011 . D 

h :. a n /3 = A . = : x e a . y e /9 . D*, y . #Jy : 

[#35-103] =:at/SG/:.Dh. Prop 

#50-56. Ha!(an£). = .a!{(at£)Al} 

r . #50-55 . Transp . #24-54 . D 

h: a !(an/3). = . ~ {a | £ CJ} . 

[#25-55] = .g(af /9)-«7. 

[#23-831. (#50-02)] = . & ! {(a | £) A 1} : D h . Prop 

#50-57. \-.Ina1R = l f\R[a = Ina J \R\'a 

Bern. 

V . #3516 .Dh./na1^ = a1InE 

[#50-5] =IfarsR 

[#35-17] =/ni2|^a (1) 

[#50-5] =aVr«^^ 

[#35-16-17-21] =/'HflP| k a (2) 

I- . (1) . (2) . D h . Prop 

#50-58. H:a > |J?G/. = . J R| k aG/. = .a > l J RpaGJ 

Dew. 

h.*50-57.Dh:/na > l J R = A. = ./nZ2| k a = A. = ./n«1 J Rra = A C 1 ) 
h . (1) . #50-41 . D h . Prop 

#50-59. h.(/r-o)"/8 = an/8 

Dew. 

I- . #37-412 . D h . (/fa)"/3 = /"(an /3) 

[#50-16] = a n £ . D h . Prop 

#50-6. K.R|(/ra) = i?ra 

Dew. 

I- . #35-23 . D h . i2 1 (I r«) = (^ I i")T« 
[#50-4] = i£ f a. Z) h.Prop 

#50-61. h.7"fa|i2 = a1i2 
Dew. 

h.*35-354.0K/raj J R = /!(a1 R ) 
[#50-4] =a^.D h.Prop 



SECTION A] IDENTITY AND DIVERSITY AS RELATIONS 339 

#5062. h : (I'M C a . D . ,R \ (I [ a) = R [*50*6 . #35-452]' 

#5063. \-iT><RCa.O.I[a\R = R [#5061 .#35-451] 

#5064. t-.R\(Ita i R) = R\mC'R) = R [#5062 .#22-42. #33161] 

#50-65. \-.I\-(D'R)\R = I\-(C'R)\R = R [#50-63. #22-42. #33-161] 

#507. h-.a'RCa.D.Rl'Il-a^R [#50-62. #43-11] 

#50-71. h:D^Ca.D.| J K < 7pa = i2 [#50-63 .#43-1 11] 

#50-72. \-.R\<(I\-C'R) = \R'(IfC'R) = R [#50-7-71] 

#50 73. r . R | </ = j R'I= R [#50-4 . *4311-111] 

#50-74. \-.R\\I = R\ 

Bern. 

b.*43-112.D\-.(R\\iyQ = R\Q\I 

[#50-4] =R\Q 

[#43-11] =R\'Q (1) 

h . (1) . #30-41 . D h . Prop 

#50-75. h.I'\\R=\R [Proof as in #50-74] 

#5076. b:P\=R\. = .P = R 

Dem. 

h . #34-27 . #30-41 .Db : P= R.D . P\ = R\ (1) 

b . #50-73 . #30-36 . D H : P\ = R\.D . P = i* (2) 

h . (1) . (2) . D h . Prop 
#50-761. h:|P = |^. = .P = iJ [Proof as in *50'76] 



22—2 



*51. UNIT CLASSES 

Summary o/*#51. 

In this number we introduce a new descriptive function i l x, meaning 
"the class of terms which are identical with x," which is the same thing as 
"the class whose only member is x." We are thus to have 

t'x = §{y = x). 

But y(y = x) = I'x. Hence we secure what we require by the following 
definition: 

#61-01. i=~I Df 

— > 

As a matter of notation, it might be thought that I would do as well as i, and 

that this definition is superfluous. But we need also the converse of this 

relation, and "Cnv'7 " is not a sufficiently convenient symbol. 

The propositions of this number are constantly used in what follows. It 
should be observed that the class whose members are x and y is i'x u t'y, the , 
class whose members are x, y, z is i'x u t'y u l'z, the class formed by adding 
x to a is a v i'x, and the class formed by taking x away from a is a — i'x. (If 
x is not a member of a, this is equal to a.) 

The distinction between x and i'x is one of the merits of Peano's symbolic 
logic, as well as of Frege's. On the basis of our theory of classes, the necessity 
for the distinction is of course obvious. But apart from this, the following 
consideration makes the necessity apparent. Let a be a class ; then the class 
whose only member is a has only one member, namely «, while a may have 
many members. Hence the class whose only member is a cannot be identical 
with a*. 

The propositions of the present number which are most used are the 
following: 

#5115. \-:y€i'x. = .y = x 

#5116. h.xet'x 

#512. h : x ea . = . t'xCa 

This proposition is useful because it enables us to replace membership of 
a class (x e a) by inclusion in the class (i'x C a). 

#51'211. h:iK~ea.E.i^fta = A 

#51221. V ;xea..=.(a.— i { x)\j i'x = ol 

* This argument is due to Frege. See his article "Kritische Beleuchtung einiger Punkte in 
E. Schroder's Vorlesungen iiber die Algebra der Logik," Archiv fur Syst: Phil., vol. I. p. 444 
(1895). 



SECTION A] UNIT CLASSES 341 

*51'222. I- : x~ e a . = . a — I'x = a 

#51*23. \- : i*x = i'y . = . y e i'x . = . x e i'y . = . x *= y 

#51-4. h:3!o.aCi'iK.E.a = fc'# 

/.& an existent class contained in a unit class must be identical with the 
unit class. From this proposition it will follow that is the only cardinal 
which is less than 1. 
#51*51. f- : a = t'x . = . x = i'a . = . x i a 

For classes, t'a has the same uses that (ix) (<f>x) has for functions; "iV 
means "the only member of a." We have 

*51-59. h : ^ fa (<f>z)} . = . -^ (ix) (<j>x) 



#5101. i = I Df 
#51*1. H : oxx . = . a = $ (y = x) 
Dem. 

b . #4*2 . (#51-01) . D I- : aix . = . alx . 

[#321] = .a = §(ylx). 

[#501] =.a = ^(2/ = «):Dl-.Prop 

#51*11. \-.i'x = §(y = x) [#30*3 . #51-1] 

#5112. h . E ! i'x [#5111 . *1421] 

#5113. h:o = t'a?. = .o = P(y = a?) [*2057-2 .#51*11] 

#51 131.' V : ai# . = . a = t'# [#51-113] 

#5114. h :. a = t^ . = : \y e a . = y . y = x [#5113 . *20'33] 

#51141. \-:.a=i'x.= :'&la:y€a.Dy.y = x:=:x€a:yea.Dy.y = x 

[#51-14. #14-122] 
#5115. H2/ei<#. = .2/ = # [#51-11 . #2033] 

#5116. I- . a; e i'x [#51-15 . #13-15] 

#51161. Y.Rli'x [#51-16 . #10-24] 

#5117. V.<l<i = Y 

Dem. 

h . # 51-1 . #20-2 . D h . {Q(y = x)} i x . 

[#10-24] D V . (ga) . aix . 

[#33131] Db.xed'i. 

[#1011] Dh.(x).x€<I'i. 

[*24-i4] d i- . an = v 

The above proposition is used in the theory of selections (#83-71). 



342 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*51"2. b : x e a . = . i'x C a 

Dem. 

h . #13191 . D H :. x e « . = : y — x . D y . y e a : 

[#5115] = :yei'x.D y ..yea: 

[*22-l] = :t'#Ca:.Dr.Prop 

The above proposition shows how to replace membership of a class by 
inclusion in a class; thus for example it gives: 

Socrates is a man . = . the class of terms identical with Socrates is included 

in the class of men. 

Before Peano and Frege, the relation of membership (e) was regarded as 
merely a particular case of the relation of inclusion (C). For this reason, the 
traditional formal logic treated such propositions as "Socrates is a man" as 
instances of the universal affirmative A, "All 8 is P," which is what we 
express by "aC/3." This involved a confusion of fundamentally different 
kinds of propositions, which greatly hindered the development and usefulness 
of symbolic logic. But by means of the above proposition (#51*2), we can 
always obtain a proposition stating an inclusion (namely "('«C«") which is 
equivalent to a given proposition stating membership of a class (namely 
"xea"). 

#51*21. \-.x~ea — i'x 
Dem. 

h . #22*33 - 35 .Db: xea — I'x.-.xea. x~e i'x . 

[#3-27] D.x~ei'x (1) 

h . (1) . Transp . *5116 . D h . Prop 

$51*211-. h : a;~e« . = . i'x r\ « = A 

Dem. 

H . #24*39 . D h :. i'x r\ a = A . = : y e i'x . D y . y r>j e a : 
[#5115] = :y = x .Dy.yr^ea: 

[#13191] =:^~ea:.DI-.Prop 

#51-22. \-:ar\i ( x = A.CLui'x~/3. = .xej3.ci=/3-i'x 

Dem. 

\- i #24-47 . D 

I- : a r» i'x = A . a vi'x = /3 . = . i'x C /3 . a = ft — i'x . 

[#51-2] =.xe/3.a = /3-i'x:D\-.Froi> 

#51-221. h : x e a . = . (a - i l x) u i'x — a 

Dem. 

V . #51-2 . D h : x e a . = . i 'x C a . 

[*22-G2] =.i'xvci=a. 

[#22 91] = . (a - i'x) w^ = «:Dh. Prop 



SECTION A] UNIT CLASSES 343 

#51222. b:x^ € a.=.a-L l x = a [*51'211 .*24313] 

#51*23. I- : i'x = i'y . = . y e i'x . = . x e i'y . = . x — y 

Dem. 

b . #20*31 . #51*15 . D 

I- :. i'x = i l y . = : z = x . = z . z = y : 

[#13183] =:x = y. (1) 

[#51*15] = :xei'y: (2) 

[(1).*1316] = :yei'x (3) 

h . (1) . (2) . (3) . D h . Prop 

#51*231. V ;i l xr\i i y = &. = .x^y 

Dem. 

V . #24-311 . D h :. i'x c\ i'y = A . = : i'x C - i'y : 
[#51*15] =:z = x.O z .z^y: 

[#13*191] = :x$y:. D H . Prop 

#51*232. Yi.zf-{i<x\jyy). = '.z = x.v .z = y [#22*34 . #51-15] 

This proposition states that a member of i'x u i'y must be either x or y, 
and vice versa, t.e. that i>'x u i'y is the class whose only members are x and 'y. 
#51*233. \-::a = i'xyji'y.O:.(z):.zea. = :z = x.v.z = y 

[#51*232 . #10*11 . #20*18] 
#51 234. h::a = i i x\Ji i y.D:.zea. D z . <f>z : = . cf>x . <f>y 

Dem. 
h . #51*233 .Dh ::.B.^ .0 :: z ea .D z . <j>z : = :. z = x .v . z = y :D Z . <j>z :. 
[#4-77] = :.(z):. z = x ."D .<f>z : z = y . D. <f>z:. 

[#1022] =:.z = x.D z .(j)z:z = y.D z .(f)z:. 

[#13191] = :.<^.<£y::.:>l-.Prop 

#51*235. h ::a = i'x\Ji'y.D :. (qz). z e a . <f>z . = :<)>x. v.#y 
Dem. 
h. #51*233.3 

h :: Hp . D :. (g.z) . z e a . $z . = : (qz) : z = x . v . z = y : §z : 
[#4-4] = : (qz) :z = x. <f>z. v.z = y.<f>z: 

[#10*42] =:(^z).z = x.^>z.v.('^z).z.= y.<f>z: 

[#13*195] =:^a?.v.0y::DI-.Prop 

#51*236. hz.zei'xv @.= :z = x.v.ze/3 [#2234 . #51*15] 
*51'237. H ::a= i'xv /3 . D :.(*):.*€a. = :z = x. v . .ze/3 
[#51-236 . #10-11 . #2018] 

#51*238. I- :: a = i'x v ft . D :. z e a . D 2 . $z : = : <£# : z e B . D z . <f>z 

Dem. 
h. #51*237 . D h ::. Hp . D :: zea. Z> z .$z: = :.z = x. v .zefi: D z .<f>z:. 
[#4 - 77] = :.(z):.z — x . D . (pzzzefi. D.<f>z:. 

[#10-22] = :. z = x . D z . <J>z : z e /3 . D z . <\>z :. 

[#13-191] =:.<f>x:zep.3 z .<f>z::.Dh. Prop 



344 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*51-239. h::o=t^w/3.D:. (32) . z e a . <f>z . = : <f>x . v . (gs) .ze/3.<f>z 
Dem. 

h. #51-237.3 

H : : Hp . D : . (gz) . z e a . <£* . = : (rz) :z = x . v .z e/3 : <f>z : 
[* 4 ' 4 ] s : (a*) :z=x.<f>z.v .ze@.<l>z: 

[#10-42] =:(a^).^ = «.^.v.ia^).0eyS.^: 

[*13'195] = :<£#. v. ( a *).* e /3.<^ -Oh. Prop 

#51*24. h:. L'yCi'xv fi . = ;y = x.v.ye/3 
Dem. 

I-.*51-236.D 

I- :: t*y C t<# w /3 . = :. z e i l y . D* : z = x . v . z e £ :. 
05115] =:.z = y.D z :z = x.v.zefi:. 

[#13*191] =:.y = x.v.yej3::D\-.Prop 

#51*25. l-:aCt^u/9.a;~ea.D.aC/3 [#51*211 . #2449] 
#51*3. h:yea.y^«. = .yea-t^ [#51*15 .#22*33*35] 

#51*31. t~ : ftl a r\ i ( x . = . i'x C a . = . a r\ i'x = i'x . = . x e a 
Dem. 

V . #2233 . #5115 .DhiQlanii'x .==. (gy) .yea.y = x. 
[#13*195] =.*ea. (1) 

[*51'2] =.^Ca. (2) 

[#22*621] = .i'x = i'xrMx (3) 

r.(l).(2).(3).DKProp 

#51*34. h:««o.E.-aC-( f «; [#51*2 . #22*81] 

*5135. r:se~ea.s.t-#C-a [#51*2 . #22-3,5] 

#51-36. h:a;~ea. = .«C-(-a! [#51-35 .#22-811] 
#51*36 is frequently used. 

#51-37. V.a = x (t'x C a) [#51-2 . #20-33] 

*514. h:g!a.«Ct-a!. = .a = t-« 
Dem. 

h . #24*5 . #51*15 . D h :. a ! a . a O V. = : (gy) .y ea:y ea.Dy .y = x : 
[#14'122] = :y ea. = y .y = x: 

[*51-ll.*20-33] =:a=i^:.Dh.Prop 

*51401. I- :.aCt^. = :a = A. v.a= i'x 
Dem. 

h . #51*4 . #5*6 . 3h:.aCi'x.D:cL = A.v .a = i'x (1) 

h . #2412 . #22-42 . D h :. a = A . v . a = i l x : Z> . a C t"# (2) 

l-.(l).(2).DI-.Prop 
This proposition shows that unit classes are the smallest existent classes. 



SECTION A] UNIT CLASSES 345 

#51*41. h : i'x «-» i'y = i'x\j l 1 z ,=.y=z 
Bern. 

Y . #202 . #1313 . D I- : y = z . D . i'x u i'y = i'x u t<* (1) 

I- . #22-58 . D h :. i'x v i'y — i'x «-• i'z . D : i'y C t'# «-» i's . t'-z G fc'a? w i'y : 
[#5116-232] "D:y = x.v.y=z:z = x.v.z = y: 

[*1316.*4-41] "D:y — x.z = x.\f.y = z: 

[#13*172.*2*621] D:y = z (2) 

I- . (1) . (2) . D r . Prop 

The two following propositions are lemmas for #51 43. 
#51-42. h :. i l x u i'y = t'z u t'w .D:x = z.y = w.v.x = w.y — z 

Dem. 
h . #51232 . D 

H :: t'# v i'y — i l zv i l w .= :.a = x .v ,a = y:= a ia = z .v .a = w :. 
[#101] D:.x = x.v.x = y: = ix = z.v.x = w:. 

[#13-15] D:.x = z.v.x = w (1) 

h . #202 . #1313 .Db : l'xv i'y= i l z u i*w . x — z . D . t'x \j i'y = i l x w i l w . 
[#51-41] D.y = w (2) 

Similarly H : i l x \j i'y = t'.z u t f w; .x = w .D .y = z (3) 

K (1) . (2) . (3) . D I- . Prop 

#51*421. h :. x = s . y = w . v . x = to . y = s : D . i e x *-> t'y = i l z w t'w [#51'41] 

*51'43. h :. t'# u i'i/ = t'^ u t'«/ . = :x = z.y = w.v.x = tu.y = z 
[#51-42-421] 

The following propositions are concerned with i, i.e. with the relation of 

the only member of a unit class to that class. If a is a unit class, i'a is its 

only member, (ix) (<f>x) and i'z ((f>z) are equal whenever either exists, and 
any proposition about the one is equivalent to the same proposition about the 
other. 

#51 '51. I- : a= i'x . = . x= t'a. = . xia 
Dem. 
h. #51-131. #31-11. D\-:a=i'x. = .x y t a (1) 

h . (1) . D h : x i a . y t a . D . a = i'x . a = i'y . 

[*51-23.*20-57-2] D.x = y (2) 

I- .(2). Exp. #10-11 .#471 .Dh:.xLa. = :xia:yia.D y .x = y: 

[#30-31] = : x = T'a (3) 

1- . (1) . (3) . D h . Prop 



346 

#51-511. I- . iH'x = x 



PROLEGOMENA TO CARDINAL ARITHMETIC 

[#51-51 — .*20-2~| 



[PART II 






x 



#51-51 — ,*14-2M8 



#51-52. h : E ! I'a . = . a = c'l'a 

#51-53. h:E!t' ( a. = .i'aea [#51'5216 . *14'2ri8] 

#51-54. h : E ! T'a . = . (gar) . a = i'x [#51-51 . #14-204] 

#51-55. h:EU ( a. = .El(ix)(xea) 
Dem. 

h . *51'54-14 .Dh:.E!('a. = : (g#) : y e a . = y . y = x : 
[#14-11] = : E ! (ix) (x e a) :. D I- . Prop 

#51-56. h : b = ?$ (<£?/) . = . £ (<£y) = i'b . = . 6 = (ix) (<f>x) 
Dem. 

h . #51:51 . D h :. b = t<y (<&/) . = : p (<&/) = i'& : (1) 

[*20-15.*51-ll] =:cj>y.= y .y^b: 

[#14-202] = : b = (ix) (<j>x) (2) 

1- . (1) . (2) . D h . Prop 

#51-57. I- : E ! ^y (<f>y) . = . t'y (<f>y) = (ix) (<f>x) . = . E ! (ix) (<j>x) 
Dem. 
h . #14-204 . #51-56 . D 1- : E ! 7'0 (<&/) . s . E ! O) (<£#) (1) 

H . #14 205 . D h : (ix) (<f>x) = i<# (£y) . = .(g6) . 6 = (ix) (<j>x) . b = Zp (<f>y) . 
[*51-56.*4-7l] = .(>&b).b = (ix)(<f>x) . 

[#14-20413] = . E ! (ix) (<f>x) (2) 

h . (1) . (2) . D K Prop 

#51-58. \-:En<a. = . y i'a = (ix)(x€a) [#51-57 . #203 .#14272] 

#51-59. I- : yjr fa (<j>z)) . = .yjr (ix) (<f>x) [#51-56 . #14*205] 



#52. THE CARDINAL NUMBER 1 

Summary of #52. 

In this number, we introduce the cardinal number 1, defined as the class 
of all unit classes. The fact that 1 so defined is a cardinal number is not 
relevant at present, and cannot of course be proved until "cardinal number'' 
has been defined. For the present, therefore, 1 is to be regarded simply as 
the class of all unit classes, unit classes being such classes as are of the form 
i'x for some x. 

Like A and V, 1 is ambiguous as to type; it means "all unit classes of 
the type in question." The symbol "1 (a)," where a is a type, will mean "all 
unit classes whose sole members belong to the type a" (cf. #65). Thus e.g. 
"£ e 1 (Indiv)" will mean "'£ is a class consisting of one individual," if "Indiv" 
stands for the class of individuals. 

The properties of 1 to be proved in the present number are what we may 
call logical as opposed to arithmetical properties, i.e. they are not concerned 
with the arithmetical operations (addition, etc.) which can be performed with 
1, but with the relations of 1 to unit classes. The arithmetical properties of 
1 will be considered later, in Part III. 

The propositions of the present number which are most used are the 
following: 
#5216. h :.ae 1 . = : 3 ! a : x,y ea. D x>y . x = y 

I.e. a is a unit class if, and only if, it is not null, and all its members are 
identical. 

#5222. f- . i'x e 1 

#524. f- :.ael v i'A . = :x, yea. "5 x>y .x= y 

We shall define as t'A. Thus the above proposition states that a class 
has one member or none when, and only when, all its members are identical. 

#52-41. h : 3 ! a . a»->6 1 . = . fax, y).x,yea.x^y 

This proposition is obtainable from #52-4 by transposition, i.e. by negating 
each side- of the equivalence. 
#52-46. h:.a,/3el.D:aC/3. = .a = /3.= .3!(an/3) 

I.e. two unit classes are identical when, and only when, one is contained 
in the other, and when and only when they have a common part. 



#52-01. 1 = a {fax) . a = i'x) Df 

#521. h:ael. = .fax).a = i'x [#20-3 . (#52-01)] 



348 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#5211. h:.a€l. = :(^w):.yea.=y.y = x [#521 . #5114] 

#5212. h:2(<^)el. = .E !(?*■)(<£«■) 

Dem. 

h . #5211 . D I- :. %(4>z)e 1 . = : (g«) :yez(<f>z) .= v .y = x: 

[*20'3] s:(3«): <f>y.= y .y = x: 

[#1411] = : E ! (7*) (<£#) :. D h . Prop 

#5213. Kl = D'i 

Bern. 

h . #51131 . D h : a = t'x . = . uix : 

[♦10-11-281] D I- : (ga?) . a = i'x . = . (-#r) . aix : 

[#521] D h : a e 1 . = . (ftx) . aix 

[#3313] = . a e ~D'i : D H . Prop 

#5214. r- . 1 = t"V [#5213 . #37-28] 

#5215. h : a e 1 . = . E ! t'a [*51'54 . #521] 

#5216. \-:.ael. = :Rla:x,y€a.'D x>y .x = y [#52*15 . #51-55 . #14*203] 

*5217. b:ael. = .~i'a = {ix)(xea) [#51-58. #52-15] 

#52171. h : a e 1 . = . E I (ix) (x e a) [#51 -55 . #52-15] 

#52172. h:ael. = .i-T'a = a [#51-52. #5215] 

#52173. h:ael.= .r-aea [#5153 .#5215] 

#5218. bz.ael .= :(Qx):xea:y€Ci.Dy.y = x 
Dem. 
h .#51-141 . D I- :.(g#).a= i'x . = : (a#):a-ea:yea . D y .y — x (1) 
h . (1) . #52-1 . D I- . Prop 

#52181. H:.a~el . = :a?6a.D a ..(33/).2/ea.y4 : « [#5218 .#1-0-51] 
#52-2. h.lCCls 
Bern. 

\~ . #52-1 . D h : a e 1 . D . (gar) ,a = l'x. 

[#51-11] D.(aa?).o = $(*•=*). 

[#20-54] D.(a^<£).2(<£!£) = 2(* = #)-« = 2(<M*)- 

[#10-5] D.(a<£).a = £(£!*). 

[*20'4] D.aeClsOI-.Prop 

#5221. h.A^el 

Dem. 

K*5216.Dh:ael.D a .g!a: 

[#2463] Dh:A~el 
#5222. h.t'ael [#51-12 . #14-28 . #10-24 . #52-1] 



SECTION A] THE CARDINAL NUMBER 1 



349 



#5223. 


V 


au.ai-i 














Bern. 




h . *52-22 . 


#10-24 . 


Dh 


■ (a*) 


. l l x e 1 . 










[*20'54] 




Dh 


■ (a*, o 


t) . a = fc'# 


ael. 








[*10'5] 




Dh 


■ (a«) 


.ael 




(1) 






V . *52'21 


. #2235 . 


Dh 


Ae- 


1. 










[*10'24] 




Dh 


• (a«)- 


ae-1 




(2) 






K.(l).(2) 


• 


Dh 


. Prop 








#5224. 


h 


1 + A n Cls . 1 


4= V n Cls [*52-23 . 


#24-54 . #2417 


Transp] 


#523. 


h 


. i"a C 1 














Bern. 




1- . #5222 . 


#2-02 . 


D h : y e a . D . i l y 


si : 








[#5M2.*1011.*37-61] D h . t"a 


CI 






*5231. 


h 


: k C 1 . = . (got) 


. k = i"a 












Dem. 



















h 



V . *52\L4 . D f- : k C 1 . = . k C t" V . 
[*37-66.*51-12] =.(aa).aCV.« = i"a. 

[#24-11] = . (ga) . « = t"a : D r . Prop 

aelw t'A . = :x,yea.. D x>y -x — y 



(1) 

(2) 



#52-4. 

Dem. 
h . #52-16 . #24-54 . D 

h :. a e. 1 . = : a =f= A : x, y e a . D X)J/ . x = y:. 

[#4*37] Dh::ael.v.a = A: = :.a = A:.v:.a=i=A:a?,yea-. D x>y .x = y:. 
[#5*63] = :. a = A : v :x, yea, D XtV .x = y 

h . #24*51 . #10-53 . #1162 . D I- :. a = A . D : x, y e a . D x , y . x — y 
V . (1) '. (2) . #4-72 . Dh::oL€l.v.a = A: = :.x,yea. 3 x>y .x = y(S) 

r- . (3) . #51-236 . DKProp 

This proposition is frequently useful. We shall define the number as 
t'A ; thus the above proposition states that a class has one member or none 
when, and only when, all its members are identical. It will be seen that 
x, y e a . "5 x>y . x = y does not imply g ! a, and therefore allows the possibility of 
a having no members. 

#52'41. I- : g ! a . a~ e 1 . = . (g#, y).x,yea.x^y 
Dem. 



h . #24-54 .Dh:.g!a.o~el.= 
[#4-56] = 

[#51-236] = 

[#52 4. Transp] = 

[#11-52] = 



a 4= A . a~e 1 : 

~{ael.v.a = A}: 

~(ael u t'A): 

~{x,yea.D x , y .x = y] 

(3^> y) ' x, y e a . x =^ y : . "D h . Prop 



350 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#5242. h:.ael.:>:g!an/3. = .an/3el 
Bern. 
h.*5131. Dh:.a!i'a?n£. = .t'a?n/8 = i'a?:. 
[*20'53] Dh:.a = t'#.D:g!an£.= .an/3=i'#:. 

[*10-11*28] D h :. (gar) . a = t'# . D : (ga?) :a!an y g. = .an y g = (^: 
[#10-37] D:a!ort / 8.D.( a a ? ).an/8 = t'ar (1) 

K(1).*52\L .Dh:.ael.D: a !an/3.D.an/3el (2) 

h. #52-16. Dh:an/3el.D.a!an/3 (3) 

h.(2).(3). I) K Prop 

#5243. h:ael.a!an/3.= .a€l.a«/Sel [#62'42 . *5'32] 
#52-44. h:.«el.D:a!any8. = .aC/8. = .an/9 = a 
Dew. 

H. #51-31. Dha!i'«nj8. = .i'*Ci9: 

[*13-13.Exp] Dh:.«=t'*.3:g!oftj8.= .oCj8:. 

[#10-11-23] Dh:. fax) .a = ^.D:g!an/3. = .aC/3:. 

[#52-1] Dh:. ael.D:g!an£.= .aC/3 (1) 

h.(l).#22621.Dh.Prop 

#52-45. h::a,y8el.D:.aC/3u7. = :a = /9.v.aC7 
Dem. 

K#51-236^^.D 

.Z, #, /3 

\- :. x e l'y u ry .= ; x = y .v . oc €<y :. 
[#51-2-23] D h :. i l x C t'y w 7 . = : t'cc = i l y . v . t<# C y :. 
[#13-21] Dh:a = ^./3 = t'y.D:.aC i 8u 7 . = :« = ^.v.aC7:: 

[#11-11-35] DH::(aaj,2/).a=i^.y8=i < y.D:.aC^w 7 .=:a= y 8.v.aC 7 (1) 
K (1). #521. Dr. Prop 

#52-46. h:.a,/3el.D:aC/3. = .a = /3. = .g!(an/3) 
Dem. 
h . #51-2-23 . Z> f- : i<x C i'y . = . i'x = i f y (1) 

h.(l). #13-21. DI-:.a=t'a>.£ = i'y.D:aC/8.=5.a = £ (2) 

b . (2) . #11-11-35 . #52-1 .DI-:.a,£el.D:aC£. = .a=»/S (3) 

I- . (3) . #52-44 . D K Prop 

#526. I- :. a e 1 . D : # e a . = . i'x — a . = . # = t'a 



Dem. 



H. #51-23. Dl-:a?€t'y. = .t'aj = i'y: 

[*13-13.Exp] D I- :. a = t'y . D : a? e a . = . i'x = a :. 
[*1011-23.#521]D(-:.ael. D : xe a . = . i'x = a . (1) 

[#51-51] = .#=T'a (2) 

h.(l).(2).DKProp 



SECTION A] THE CARDINAL NUMBER 1 351 

#52*601. I- :: a e 1 . D :. <f> (i'a) . = : x e a . D x . <j>x : = : (g#) .xea.<j>x 
Dem. 

b . #5215 . D h :. Hp . D : E ! T'a : (1) 

[*30 4] D : x i a . = . a? = i'a . 

[*52-6] = .xea (2) 

h.(l).*30-33.D 

h :: Hp . D :. </> (t'a) . = : x i a . D^ . <f>x : = : fax) ,xia.<j>x (3) 

I- . (2) . (3) . D h . Prop 
#52*602. H :. 2 (^) e 1 . D : >|r (7#) (<f>x) . = . <f>x"D x yjrx . = . fax) . <f>x . yfrx 

[#5212 . #14-26] 

#52-61. h^ael.Dr^ae^.s.aC/S.^.glCan/S) [#52-601 ^? 

L V x - 

#5262. h:,«,£el.D :«=£. = . *'«=*'£ 

V . #52*601 . D I- :: Hp . D :. 7'a= ?£ . = : xea . D z .x = ~i<$ : 
[#52 - 6] = :x€a.D x .X€/3 : 

[#52*46] = : a = : : D I- . Prop 

#5263. h:a,/Sel.o + /9.D.an/8 = A [*52'46 . Transp] 

#52*64. h:ael.D.ar»/3elut«A 

Dem. 

h . #52*43 . DhrHp-glan^.D-anySel: 

[*5*6.*24*54] D h :. Hp . D : a n /3 = A . v .'a n # e 1 : 
[#51-236] D : a n £ e 1 u t'A :. D h . Prop 
#527. h:./3-a6l.aCf.fC/3.D:^=a.v.| = ^ 
Dem. 
K #22-41 . DH:Hp.fCa.D.| = a (1) 

K #24*55. Dh:~(fCa).D.a!^-a (2) 

K #22-48. Dh:Hp. D.f-aC/S-a (3) 

K(2).(3). Dh:Hp.~(£Ca).I>.a!£-a.£- a C/3-a (4) 

I- . #52-1 . D I- : Hp . D . fax) .0-a = i*x (5) 

K(4).(5).*51-4.Dr:Hp.~(fCa).:).f-a=/3-a. 
[#24-411] 3-f = £ (6) 

+ .(l).(6).Dh.Prop 



*53. MISCELLANEOUS PROPOSITIONS 
INVOLVING UNIT CLASSES 

Summary o/#53. 

The propositions to be given in this number are mostly such as would 
have come more naturally at an earlier stage, but could not be given sooner 
because they involved unit classes. It is to be observed that i'x \j I'y is the 
class consisting of the members x and y, while i'x f i'y is the relation which 
holds only between x and y. If a and (3 are classes, t'a u t'/3 is a class of 
classes, its members being a and #. If R and S are relations, i'R 7 l'S is a 
relation of relations; and so on. 

The present number begins by connecting products and sums p'te, s'k, 
p'X, s'\, in cases where the members of k or A. are specified, with the products 
or sums a n /3, a u /3. R n 8, R vy S. We have 

*53'01. \- • p'i'a = a 
#631. K|)'(i'«w^)=oft j 8 
#5314. H . p'(/c v 1*0) = p'k r\ a 
with similar propositions for s, p and s. 

We have next a set of propositions on sums and products of classes of unit 
classes. The most important of these is 
#53 22. h . s'i"a = a 

We have next a proposition showing that the sum of k is null when, and 
only when, k is either null or has the null-class for its only member, i.e. 
#5324. 1- :. s'k = A . = : « = A n Cis . v . k = t'A 

(Here we write "A r\ Cls," to show that the "A" in question is of the next 
type above that of the other two A's.) 

We have next various propositions on the relations of R l x and R'x and 
R"a in various cases, first for a general relation R, and then for the particular 
relation s defined in *4<0. Three of these propositions are very frequently 
used, namely: 
#533. I- : E ! R'x . = . R'x e 1 

#53 301. h . R"i'x = R'x 

— > 
#5331. h : E ! R'x . D . R"i'x = t'R'x = R'x 

The remaining propositions of this number are of less importance, and are 
seldom referred to. 



SECTION A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES 353 

#5301. \-.p'i t z = a 

Dem. 

h . #401 . D I- :. x ep'i'a . = : fi e t'a . Dp . x e £ : 

[*51-15] =:j3 = a.D fi .xe]3: 

[#13191] = :xea:. D h . Prop 

#5302. h.sV«=a 

Dem. 

h . #4011 . D h : x e s't'a . = . (ftp) . £ e i'o. . x e /3 . 

[*51'15] =.( a /3).£ = a.#e/3. 

[#13*195] = . x e a : D h . Prop 

#6303. H . p'i'R = R [Proof as in #5301] 
#5304. h . s'l'R - 22 [Proof as in *53'02] 
#631. Kp'(t < «ut' i 8) = fln J 8 

I- . #4018 . D ' h . p.'(t'a w t<£) = j/t'a n p'i<£ 
[#53-01] =any8.Dh.Prop 

This proposition can be extended to t'a v t'yg u I'y, etc. It shows the 
connection (for finite classes of classes) between the product p l ic and the 
product of the members a r\ (3 r\ 7 n .... 

#5311. h.s'(i'aot</3) = a v£ 

Dem. 

h . #40171 . D H . s'0'a u t<£) = sVa u s't'0 

[#5302] =au/3.Dh.Prop 

Similar remarks apply to this proposition as to #53*1. 
#5312. h . p'{i'R \Ji'S) = RhS [#41-18 . #5303] 

This proposition shows the connection between the product p*/c for a class 
k consisting of two relations R and 8, and the product Rf\S. The proposition 
can be extended to the product of any given finite class of relations. 
#6313. \-.s'(i'Rvji'S) = RvS [#41-171 .#5304] 

Similar remarks apply to this proposition as to #53'12. 
#6314. h . p'{ic v 1*0) =p*K na 

Dem. 

F- . #40-18 . D h . p'( K v 1'a) =p' K n p'l'a 
[#5301] =p<K n a 

#5315. l-.*'(Kui'«) = ^ua [Proof as in #5314] 

#5316. h . p'(\ v i'R) =p'\nR [Proof as in #5314] 

#5317. f . s'(\ v i'R) = s'\ c/ R [Proof as in #5314] 

The above proposition and the next are both used in connection with 
mathematical induction (#91*55 and #97'46 respectively). 

R&W I 23 



354 PEOLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#5318. h.s<(a-t<A) = s'a 

Dem. 

h . #51-221 . D H : A e a . D . (a - t'A) u t'A = a . 

[#5315] D.s'(a-i'A)vA = s'a. 

[#24-24] 3 . s'(a - i l A) = s'a (1) 

h . *51 222 . D h : A~ea. D. a-f/A=a. 

[#30-37] D.*'(a-i'A) = *'a (2) 

h.(l).(2).D>.Prop 

*53181. I" . s'(\ - t'A) = s'X [Proof as in #53-18] 

#532. I- : /eel .D. t i K=p i K = s t K 

This proposition requires, for significance, that k should be a class of 
classes. It is used in #88*47, in the number on the existence of selections 
and the multiplicative axiom. 

Dem. 
V . #52-601 . D H :: Hp . D :. # e t '* : = : a e * . D tt . «e«: = : (go) . a e /c . x e a (1) 
h . (1) . #40 Ml . D h . Prop 

*5321. h : X e 1 . D . T'X = jp'\ = s'\ [Similar proof] 

This proposition requires, for significance, that X should be a class of 
relations. 

#5322. h . s' i"a = a 

Dem. 

H . #4011 . D h : x e s'l"a . = . (37) . 7 e i"a .#67. 

[*37-64.*5112] = . (33/) .yea.xei'y. 

[*5115] = .(^y).yea.x = y. 

[#13-195] =.^ea:Dh.Prop 

#53221. \-.i"(i'xyJL'y) = i t i'xvL'i'y 
Dem. 

K #371 . D f- :. a e t"(i'# u i'y) . = : (g^) . z e (t'x v i l y) .aiz: 
[#51131] = : (a*) • z e (t'« w t'y) . a = i'z : 

[*51'235] = :a = i'x.v.a=i'y: 

[#51-232] =:ae (t'l'x u tVt/) :. D h . Prop 

#53 222. \-:k = i"a . D . a = fc "* 
Dem. 

h . #13-12 . #202 . D r : Hp . D . t "« = t "i"a 

[#51-511.*14-21.*37-67] = £ {(32/) .yeci.x^i H l y\ 

[#51-511] = & {(32/) ■ V € a . x = y] 

[#13195] = a:DKProp 



SECTION A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES 



355 



*53'23. h:/eCl.D.s'« = t"K 

Dem. 

b . *52-31 . D b : Hp . = . (ga) . « = t"a 

b . *53-22 . D b : k = i"a . D . s'/c = a 



(1) 

(2) 



j>53'222] = i"* 

r- . (1) . (2) . *10-ll-23 . D f- . Prop 

$53*231. h :. # e a . D,,, . # = ;?/ : = : a = A . v . a = t'y 

Dem. 

b . *51141 . D I- :. g ! a : x e a . D x . x = t/ : = : a = t'y (1) 

I- . *10'53 . D b :.~g I a . D : x € a . D x . x = y :. 
[*4'71] D b :.~g !a:#ea. D^ . # = ?/: = .~[>{ ! a . 

[*24-51] = .a=A (2) 

b . (1) . (2) . *4-42'39 . D h . Prop 

*53 24. h :. s'k = A . = : k = A n Cls . v . * = t'A 

Dem. 

b . *2415 . *4011 . D 

b :.s'k = A . = : (x) :~{(aa) . ae k . xea] : 

[#10-51] =:(«,a):«e«.D.a~eK: 

[*112.*10-23] = : (g#) .#ea.D a .a~e«: 

[*24-54] =:a4=A.D a .a~e«: 

[Transp] = : a. e k . D a . a = A : 

[*53231] = : * = A n Cls . v . K = t'A :. D b . Prop 

In the enunciation and the last line of the proof of the above proposition, 
we write "k — A n Cls" rather than "k = A," because this A must be of the type 
next above that of the A in "k = t'A." 

The following proposition is used in the theory of selections (#83 - 731). 

*53 25. I- :. s'k n s'\ = A . D : k r\ A. = A n Cls . v . k n \ = t'A 
Dem. 
b . *40 181 . D I- :. Hp . D : s'(k n X) = A : 
[*53 24] D : k r> \ = A n Cls . v . k n \ = t'A :. D b . Prop 

*53 3. H:E!i2^. = . J R^el 

Dem. 

(%b):yRx.=y.y = b: 



b . *302 . D I- :. E ! R ( x . = 
[*32-18.*5115] ee 

[*2031] = 

[*52-l] = 



fab)iyeR'x.= y .yel'b 
(Rb).R'x = i'b: 
i2^el:.Dh.Prop 



The above proposition is very frequently used. 



23—2 



356 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



*53301. \-.R«i i x = R'x 

Dem. 

V . #371 . #5115 . D r : y e R'H'x . = . (g«) .z = as. yRz . 

[#13-195] = .yRx. 

[#32-18] = . y eR'x : D h . Prop 

#53302. h.R"(i'xvi'y)=~R'xvR'y [#37-22 .#53301] 

The above proposition is used in the cardinal theory of exponentiation 
(#116-71). 

#53-31. h: El R'x.D.R"i'x = t'R'x = R'x 

The above proposition is one of which the subsequent use is frequent. 

Dem. 

h . #51-11 . #1418 . D h : Hp . D . i'R'x = §(y = R ( x) 

[#30-4] =§(yRx) 

[#32-13] = R'x (1) 

h . (1) . #53-301 . D r- . Prop 

#53-32. I- : E ! E'a: . E ! R'y . D . R"(i'x w I'y) = i'R'x u t'iZ'y 

r . #37-22 . D H . E"(t'a: v, t'</) = U"t'a? u R"i'y (1) 

h. (1). #5331. Dr. Prop 



#53-33. h . s"l'k = t V/e #53-31 



#5334. h . s"{i'k u i'X) = t's'/c v i VX #53-32 



i2 



ij 



#53-35. h . a'*"(i'« w t'X) = s'* « s'X = s'(k \j \) 

Dem. 

r- . #53-34 . D I- . sV'(i'« w i'X) = s'(t's'fc vj l's'X) 

[#53-11] =«'«us'X 

[#40-171] = s'(k u X) . D I- . Prop 

The above proposition may also be proved as folloAVs: 
h . #42-1 . D h . s's"(i'ic v t'X) = *V(t'«c v i'*.) 
[#53-11] = s<O w >-) 

[#40-171] =*'«u*'\.Dh. Prop 

— -► — » — ► ^ — > 

#53-4. H : a? = iZ'y . = . R'y e 1 . a; e J2'y . = . t'a; = fi'y . = . x = t'B'y 

h . #14-21 . #4-71 . D I- : a; = R'y . = , E ! R'y . a? = R'y . 
[*30-4.*5-32] = . E! R'y. xRy . 

[*53-3.*32-18] = . R'y e 1 . x e R'y . 

[*52-6.#532] = . R'y el.i'x = R'y . 



(1) 



SECTION A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES 357 

— ► 

[*52-22] = . v'x = R'y . (2) 

[*51-51] =.x=SR'y (3) 

I- . (1) . (2) . (3) .. Dr. Prop 
*53*5. I- : a ! a . = . a e Cls - fc'A 

Dem. 

h . *20-41 . D h : a ! z (<j>z) . = .z (<f>z) e Cls . a ! 2 (<^) . 
[*24-54] = . z (<j>z) eCls.z (<f>z) + A . 

[*51-3] =. 2 (<j>z) e Cls -i'A:DK Prop 

In the above proof, as usually where "Cls" or other type-symbols occur, 
it is necessary to abandon the notation by Greek letters and revert to the 
explicit notation. 

*53 51. I- : a ! R . = . R e Rel - i'A [Proof as in *53"5] 
*53 52. \- :aefc .Rla. = .a€ tc- i'A 

Dem. 

h . *2454 . D r : a e k . a ! a . = . a e k . a 4= A . 

[*51'3] = .ae«-i'A: DH . Prop 

*53 53. V-.ReX.&XR.^.RizX-i'h [Proof as in *53'52] 

The following propositions are inserted because of their connection with 

the definition of a -* /3 in *70. R"G.'R and iJ"V are both important classes. 

*536. \-:R = A.Rla.'}.~R"a=i i A.R"a = i'A 
Dem. 

h . *33*1 5241 . *2413 . D r : Hp . D . R'x = A (1) 

h . (1) . *37-7 . D r : Hp . D . #"a = £ {(gar) . a? e a . £ = A} 
[*10'35] = /§ {g ! a . £ = A} 

[*4'73] =i §(/3 = A) 

[*51-11] = t'A (2) 

Similarly r : Hp . D .#"« = t'A (3) 

h . (2) . (3) . D h . Prop 

*53'601. h : a ! a . a n a'.R = A . D . jR"a= t'A 

h . *3341 . D h : Hp . x e a . D . i2'a > = A (1) 

h . (1) . *37-7 . D h : Hp . D . R"a = § {(ga?) . a? e a . /3 = A} 

[*10-35] = /§ (a ! « . £ = A} 

[*4-73.*5111] = l'A : D h . Prop 

*53 602. h:g!a.an D<12 = A . D . i2"a = t'A [Proof as in *53601] 
*53 603. \-:'&l-a<R.D.ll"(-a'R) = i' A [*24'21 . #53-601] 



358 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#53 604. r : a ! - D'R . D . R"(- D'R) = i'A [#2421 . #53-602] 

#53 61. r : CL'R C a . d'R =f a . D . ~R"a = #"(I<# u i'A 

Dew. 

h.#22'92. D\-:Hp.D.a = a<Rvj(a-(I'R) (1) 

K#24-6. . DHHp.D.gla-CKR. 

[#24-21. #53-601] D.~R"(a-d'R) = t'A (2) 

K (1) . #3722 . D I- : Hp. D . R"<i=~R''d'R v R"{a-d'R) 
[(2)] = £"(KR ui'A-.Dl-. Prop 

#53-611. h : D'i2 C a . D'R + a . D . S"a = E"D'ir: u t<A [Proof as in #53-61] 
#53612. h-.d'R^V .D.R"V = R'"a ( RuL'A [#53-61 .#2411] 

#53 613. r : D'R =J= V . D . S~"V = S"D<£ u t'A [#53'611 . #2411] 

#53-614. \-.ll"d'R = R"V-i'A 

Dem. 

V . #53-612 . #22-68 . #24*21 . D 

V : d'R 4= V . D . J?"V - ('A = Jfi?<<Pis: - t'A (1) 
f- . #22-481 . D f- : <!<£ = V . D . £"V - i'A = R"d ( R - i'A (2) 

h . #37-772 . #51-36 . #22-621 . D I- . R"d'R - i'A = R"d'R (3) 

h . (1) . (2) . (3) . D V . Prop 

#53-615. I- . R"D'R = R"V - i'A [Proof as in *53'614] 
The two following propositions are used in #70"12. 

#53 62. h : R ii a i R C y . = . ll"Y C y u i< A 
Dem. 

\- . #53-614 .Dh 2S"(I«fl C 7 . = . R"V - i'A C 7 . 

[#24-43] = . R t( V C 7 v i'A : D h . Prop 

#53621. h : £"D'i2 C 7 . = . i£"V C 7 w('A [Proof as in #53-62] 
#5363. h :<1'R$V .3 .D'R = R*'<l<Ryj i'A [#3778 .#53-612] 
#53-631. h :D'R$V .0 .D'R = %'D'Rv i'A [#37781 .*53'613] 
#53-64. h : d'R = V . D . D'j?= 5"<2<£ [#37-78] 

#53641. r : D'i2 = V . D . D^ =*R"D'R [#37-781] 



*54. CARDINAL COUPLES 

Summary o/#54. 

Couples are of two kinds, namely (1) i'x w t% in which there is no order 
as between x and y, and (2) i'x f *>% ™ which there is an order. We may 
distinguish these two kinds of couples as cardinal and ordinal respectively, 
since (as will be shown hereafter) the class of all couples of the form i'x u i'y 
(where x 4= y) is the cardinal number 2, while the class of all couples of the 
form i'x t i'y (where x^y) is the ordinal number 2, to which, for the sake of 
distinction, we assign the symbol "2 r ," where the suffix "r" stands for 
"relational," because the ordinal 2 is a class of relations. In the present and 
the following numbers, we shall define 2 and 2 r as the classes of cardinal and 
ordinal couples respectively, leaving.it to a later stage to show that 2 and %, 
so defined, are respectively a cardinal and an ordinal number. An ordinal 
couple will also be called an ordered couple or a couple with sense. Thus a 
couple with sense is a couple of which one comes first and the other second. 

We introduce here the cardinal number 0, defined as i'A. That so 
defined is a cardinal number, will be proved at a later stage ; for the present, 

we postpone the proof that so defined has the arithmetical properties of 

zero. 

Cardinal couples are much less important, even in cardinal arithmetic, 

than ordinal couples, which will be considered in the two following numbers 

(*55 and'*56). It is necessary, however, to prove some of the properties of 

cardinal couples, and this will be done in the present number. Some properties 

of cardinal couples which have been already proved are here repeated for 

convenience of reference. The definitions of and 2 are: 

*54 01. = i'A Df 

*5402. 2 = oi{(>&x,y).x$y.a=i'xyJL'y} Df 

Most of the propositions of the present number, except those that merely 

embody the definitions (*541-10ri02), are used very seldom. The following 

are among the most important. 

*5426. ^ : i'x «-» i'y e 2 . = . x^y 

*54'3. h .2 = a{('&x).xea.a-i t x6l} 

*54-4. \-:./3Ct'xvJt'y.= :p = A.v.j3 = i t x.v.j3=i'y.v.8 = t'xyji,'y 

*54*53. h : a e 2 . x, y e a . x ^ y . D . a = i'x u i'y 

*54-56. h:«~e0ulw2. = . (g#, y,z) .x,y,zea.x^y .x^z .y^ z 



360 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#54-01. = i'A Df 

#5402. 2 = o {(ga>, y).x^y.a = i'x\j i'y) Df 

#541. h . = i'A [(#64-01)] 

#54101. h s a e 2 . = . (gar, y) .x^y .u= i'x\j i'y [(#5402)] 

#54102. l-:aeO. = .a = A [#541] 

The two following propositions have already occurred in #51, but are here 
repeated, because they belong to the subject of the present number. 

#54-21. h : i l x u i l y = i'x\j i'z . = .y = z [#51-41] 

#54*22. \*:.i'xv i'y=i'z\j i'w. = :x = z .y = w.v ,x = w.y = z [#51-43] 
*5425. h : i'x u i'y e 1 . = . x = y 

Bern, 
b . #52-461 . #22-58 . D I- : i'x u i'y e 1 . D . i'x w i'y = i'x . i'x w i l y = i'y . 
[#20-23] D. i'x = i'y (1) 

r- . #22-56 . Dh:i'x = i'y.D.i'x\j i'y = i'x . 

[#52-22] D . i'x u t'y e 1 (2) 

I- . (1) . (2) . D h : i'x u i'y el . = .l'x = i'y . 
[#51-23] = .ar = y: Dr. Prop 

#54-26. I- : i'x v i'y e 2 . s . a; =j=y 

Dew. 
I- . #54-101 . D h :: t'a; v i'y e 2 . 

= :. (gz, w) . z 4= w . t'# u t'y = 1*2 u i'w :. 

[#54 - 22] ■= :.(g2, w) :z^w:x = z.y = w. v .x = w.y = z:. 
[*4-4.#ll-41] = :. (gs, w).2r + w.a; = ^.3/ = M;.v. (gs, w) . z=^w ,x = w ,y = z :. 
[#13-22] =:.a? + y.v.y + a?:. 

[#1316] = :.#=|=y:OI-.Prop 

#5427. r . t<# w^elu2 [#54-25-26] 

#54-271. h . 1 w 2 = a {(gar, y).a = i'x\j i'y] 

Dem. 
K*4-42.D 

^•••ct=l'xyji'y. = :x = y.a=i'xvi'y.v.x^y.a=i'x\Ji'y (1) 

I- . (1) . *1111'341-41 . D h :. ( aaJ> y) . a= l<* u i'y . 

= : (a«» y).* = y.o = t'awt'y.v. (gas, y) . «+y . a = t<# v t'y : 
[#13195] = : (ga?) . = t'a; u t'a; . v . (g#, y) . a- 4= y . a = t'# w i<y : 

[#22-56] = : (ga;) . a = i'x . v . (ga;, y) . x^y .a= i'xu i'y : 

[*52-l.*54101] =:ael.v.ae2: 
[#22-34] = : a e 1 w 2 :. D h . Prop 



SECTION A] CARDINAL COUPLES 361 

*543. f- . 2 = a {(ga?) . x e a . a - i'x e 1} 

.Dew.. 
I- . *52 1 . *1035 . D 

h : (g#) .xea.a-i'xel. = . (g«, y) • # e a . a - i'x = i'y . 

*51-22 ^l?1 = . (3«, y) • i'x rs i'y= A . i l x v i'y = a . 

«,0] 

[*51-231.*54*101] = . a e 2 : D h . Prop 

*544. \- :. C i'x v i'y . = : = A .v . = i'x . v . = i'y .v . = i'x u i'y 

Dem. 
h.*51-2. D\-:x,ye0.D.i'xyJl'yC0: 

[Fact] D \- : 0C i'x v i'y . x,y e .D . C i'x \j i'y . i'x v i'y C . 
[*2241] D.@=i'xsji'y (1) 

h . *51'25 .3\-:.0Ci'x\Ji'y.y~€0.O:0Ci'x: 

[*51'401] D : /3 = A . v . £ = i'# (2) 

Similarly h :.0Ci'x \j i'y .x~e .2 : = A.v . = i'y (3) 

h . (2) . (3) . *3-48 . D 

I- :. C i'a? u i'y . ~(x,y € 0) . D : = A . v . = i'x . v . = i'y (4) 

I- . (1) . (4) . *34-8 . D 

\-:.0Ci'xvi'y.O:0 = A.v.0 = i'x.v.0=l'y.v.0 = l'xyJi'y (5) 

I- . *2412 . *22-58-42 . D 

h :. = A . v . = i'x . v . = i'y . v . = i'x u i'y : D . C I'a? w> i'y (6) 

K (5). (6). Dr. Prop 

This proposition shows that a class contained in a couple is either the 
null-class or a unit class or the couple itself, whence it will follow that and 
1 are the only numbers which are less than 2. 

*54'41. r-::cte2.D:.£Ca.3:£=A.v'./9el.v.£62 

Dem. 
h.*52\L. D\-:.0 = i'x.v.0=i'y:D.0el (1) 

V . *54-26 . D h :. x + y . D : = i'x u i'y . D . £ e 2 (2) 

K (1) . (2) . *54-4 . D 

H : : x 4= y . D : . £ C I '# w i 'y . D : /9 = A . v . e 1 . v . e 2 : : 
[*13-12] 3 h :: a = i'x v i'y . x^y .0 :. C a .D : = A .v . el .v . e2 :: 
[1111-35] D 

h :. (g#,y) . a = i'# v i'y. x^y.D:.0Ca:0 = A.v.0el.v.0e2 (3) 
h . (3) . *54101 . D h . Prop 

*54-411. h:.ae2.D:/3Ca.D./9e0wlw2 [*5441102] 



362 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#5442. K::ae2.D:./3Ca.g!/8./34=a. = ./3et"a 

Dem. 
K*54-4. Dh:« = i^ut^.D:. 

/3 C a . g ! /3 . = : £ = A . v . £ = t<# . v . £ = ty. v . £ = a : 3 ! /3 : 
[*24-53-56.*51-161] = :/3=t f #. v ./3= i'y . v .£ = « (1) 

f- . #5425 . Transp . #5222 . D h : a? + y . D . l'a> u t'y + t'a> . t'a; v t'y + t 'y : 
[#13-12] Dh:a = t'aut'y.a!^.D.a + t ( a!.a + t'y (2) 

J- . (1) . (2) . D h :: o = t'« u t'y . a; + y . D :. 



/3 = i'x . v . ft = i'y 
(gf) . z ea.fi = i'z 
6 t"a (3) 



£ C « . g ! . £ 4= a . 
[#51-235] 
[#37-6] 
I- . (3) . *ll-ll-35 . #54101 . D h . Prop 

#5443. H:.a,/3el.D:an/3 = A. = .aw / Se2 
Dem. 

h . #54-26 . D r :. a = i'/c . £ = i'y . D : o w /3 e 2 . = . a- =|= y . 
[#51-231] =.t'*nt'y = A. 

[#1312] =.on/3 = A (1) 

r.(l).*llll-35.D 

r-:.(g#,y).a = t<tf./3=t'y.D:av/3e2. = .an/3=A (2) 

h . (2) . #11-54 . #52-1 .Dr. Prop 
From this proposition it will follow, when arithmetical addition has been 
defined, that 1 + 1 = 2. 

#5444. r- : . z, w e i l x v i'y . D z tW . <j> (z, w) : = . (x, x) . cf> (x, y) . <j> (y, x) . <f> (y, y) 
Dem. 

V .#51-234. #11-62. Dr :.z, we i'x u t'y . "5 ZtW .$(z,w): = : 

zei'xv i'y . D z . <j> (z, x) . <f> (z, y) : 
[*51'234.*10-29] = :<f,(x,x).<f> (x, y>. (j> (y, x) . <f> (y, y) :. D I- . Prop 

#54-441. h ::z,w ei'xv i'y .z^iu . D z , w . <f>(z, w) :=:.x — y : v : <f>(x,y) . <£(y, x) 

Dem. 
V . *5'6 . D b :: z, w e i'x \j i'y . z =f w . D z>w . <f> (z, w) : = 
z,we t'x u i'y . D 2>w : z = w . v .<\>(z,w) 
[#5444] = : x = x . v . <f> (x, x) : x = y . v . <f> (x,y) 

y = x.v.<f>(y,x):y = y.v.<f>(y f ij): 
[#13'15] = :x = y.v.<f>(x,y):y = x.v.<f>(y,x): 

[*13-16.*4-41] = : x = y . v . <f> (x, y) . <f> (y, x) 

This proposition is used in #16342, in the theory of relations of mutually 
exclusive relations. 

#54*442. t- : : x^y ,D:.z,w e i l xu i l y. z^tv .D z>w . <f>(z,w): = ,<f)(x,y) . <f> (y, x) 

[#54-441] 



SECTION a] cardinal couples 363 

*54443. \- :: x =|= y : <f> (x, y) . = . <f> (y, x) : D :. 

z, w e i'x yj I'y . z =f w . D z , w . <j> (z, w) : = . 4> (as, y) [#54-442] 

#54*45. h :. (32, w).z,we i l x v i'y .<f> (z, w) . 

= :<f>(x,x).v.<f>(x,y).v.<f>(y,x).v.<f>(y,y) [#51-235] 

*54451. h :: ~ </> (x, x) . ~ </> (y, y) . D :. (a*, ti/) .^roet'au i'y . </> (*, w) . 

= :(f>(x,y).v.<f>(y,x) [#54-45] 

#54 452. h :: ~ <$> (x, x) . ~ <£ (y, y) : <£ (x, y) . = . <f> (y, x) : D : 

(Rz,w).z,we i'xv I'y .(f>(z,w). = .<p(x,y) [#54-451] 

#5446. V:{<&z,w).z i wei i xvi i y.z^w. = .x^y [*54'452 . #13-1516] 
#545. h :.ae2 . D :aC6 w i'w . = . a^i'zu i'iv 

Dem. 
h . *54-4 . D 

\- i.aCi'z v i'w . D : a — A . v . a = t'z . v . a == t'w . v . = i'z\j i'w (1) 

I- . *54'3 . #2454 . DHHp.D.a + A (2) 

h. #54-26 — .#1315. Dh-Hp.D.a + i'* (3) 

x,y 

K(3)-. Dh:Hp.D.o + t'w (4) 

h . (1) . (2) . (3) . (4) . *2-53 . D V :. Hp . D : a C t's u i'w . D . a = i'z u i'w (5) 
I- . #22-42 . D\-:a = i'zu L'w.D.aCi'zv i'w (6) 

h . (5) . (6) . D t- . Prop 

*5451. l-:.ae2./3elv,2.D:aC/3. = .a = /3 

Devi. 

h.*54-5.DI-:.ae2./3=i^ut < w.D:aC/3. = .a = /8 (1) 

h.(l).*ll-ll-35-45.D 

h :. a e 2 : (3*, w).^=i'2ut'w:D:«C^.E.a = 1 8 (2) 

h. (2). #54-27 l.DK Prop 
#5452. h:.a,/3e2.D:aC/3. = .a = /3. = ./3Ca [#54-51] 
*54'53. h : a e 2 . a;, y e o . &• =f= 2/ ■ 3 ■ a = i'# *»» t'y 

Dem. 

h.*51-2. Dh:Hp.D.^Co.i'!/Ca. 

[#22-59] D . i'x yj i'y C a (1) 

K #54-26. Dh:Hp. D.t'aut'ye2 (2) 

h . (1) . (2) . #54-52 . D f- . Prop 

#54-531. h :. a e 2 . D : x, y e a . x =|= y . = . a = t'# w i'y 
Dem. 

h . #54*53 . Exp . D H :. a e 2 . D : #, y e a . a; =j= y . D . a = i'# «-» i'y (1) 
I-. #54-26. D\-:.ae2.D:a=i f xuL'y.D.x^y (2) 

h. #5116. D \- : a = i'xv i'y .D . x,y ea (3) 

h.(2).(3). Dh:.ae2.D:a = t^wi < y.D.^yea. ; r=fy (4) 

K(l).(4). Dh.Prop 



364 



PBOLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#54'54. h :.ae2. = :x, yea.x^y, x , y .a = i i x\j t'y : fax, y).x,yea.x^y 

Dem. 

K #54-531. #11-11-3. Dh:.ae2.0:x,yea.x^y.0 Xty .a=i t xyj(, t y (1) 

K*5ri6.*54-101 . Dh:ae2.D.fax,y).x,yea.x^y (2) 

1- . #5-3 . #3*27 . Dh:.x,yea.x^y.D.a = i'xvi'y: D : 

x,y ea . x^y .^) .x^y .a — t'x\j ^y x. 

[*1 1-1 1-32-34] D h :. x, y e a . x^y . D x>y . a = t'x u t'y : D : 

fax,y).x,yea.x^y.D..fax,y).x$y.CL=i'xvi'y (3) 

H . (3) . Imp . #54-101 .OV :.x,y eu.x^y . D XjV . a = i'xv t'y : 

(ftx,y).x ) y€CC.xJry:D.ae2 (4) 
h.(l).(2).(4). Dh.Prop 

In the above proposition, " x, y e a . x 4= y • O x , y ■ a = ^ x w t'y" secures that 
a has not more than two members, while "fax, y) . x, y e a . x^y" secures 
that a has not fewer than two members. 

*5455. \- . u I v 2 = & {x, y e a . x ^ y . D x , y . a = i'x v t'y} 

Dem. 
I- . #4*42 . D h ::#, yea.x^y. D x<y . a = l'x v i'y : = :. 

x, y € a . x ={= y • Ox, y • a = i'® « t'y : ~ fax, y) . x, y e a . x + y :. 
v:.a;,y€0.a; + y.D (8 ,y.o = i'a?ut'y:(aaf,y).a?,yeo.a? + y (1) 
h. #11-63. D f- :. ~fax,y).x,yea .x^y.D :x,yeot .x^y. "5 x , y . a = i l x\j t'y :. 
[#4-71] D h :.x,yea .x^y. D XiV . a = t'# v i f y\ ~fax,y) .x,y€a.x^y: = : 

~fax,y)-x,ye*-n : ¥y' 

[#11-521] =:x,y ea.D Xt y.x = y : 

[#52-4] =:ae0wl (2) 

h . (1) . (2) . #54-54 . D 

h :.x,yea..x^y . D x>y . a = t'x u t'y : = : a € w 1 . v . a e 2 : 

[#22-34] =:ae0ulu2:. Dh.Prop 

#54'56. H:a~e0ulv2. = . fax, y,z) .x,y,zea.x^y .x^z .y^z 

Dem. 
K #54-55. #11 -52. D 

l-:.o~€0ulw2. = : (g#, y) . #, yea.a;^^' 01 ^ l<iZ! w l< 2/ : 
[*51-2.*22-59] = : fax, y) . t'x \j t'y C a . x =f= y . a 4= t'# « t'y : 

[#24 - 6] = : fax, y) . t'x u t'y C a . x =f y . g ! a — (t'# o t'y) : 

[#51232.Transp] = : fax, y) : i l x kj I'y C a . x ^ y : faz) .zea.z^x.z^y: 
[#51-2.#22"59] = : fax, y,z) .x, y, zea.x^y.x^Ziy^zz.Oh. Prop 

In virtue of this proposition, a class which is neither null nor a unit class 
nor a couple contains at least three distinct members. Hence it will follow 
that any cardinal number other than or 1 or 2 is equal to or greater than 3. 
The above proposition is used in #10443, which is an existence-theorem of 
considerable importance in cardinal arithmetic. 



SECTION a] cardinal couples 365 

*54'6. I" :.a n ft = A.x,x ea .y,y' eft . D: 

i'x \j i'y = i'x u i f y' . = .x = x'.y = y' 
Dem. 

\- . *51-2 . D h :. Hp . D : i'x C a . tV C a . ,i'y C ft . i l y' Cft.ar\ft = &: 
[*24*48] D : i'x u t'y = t 'a?' u t'*/' . = . i'x = t V . t'y = I'tf . 

[*51*23] =.x = x'.y = y':.1\-.Froip 

The above proposition is useful in dealing with sets of couples formed of 
one member of a class a and one member of a class ft, where a and ft have no 
members in common. It is used in the theory of cardinal multiplication 
(*113148). 



*55. ORDINAL COUPLES 

Summary q/"#55. 

Ordinal couples, which are now to be considered, are much more important, 
even in cardinal arithmetic, than cardinal couples. Their properties are in 
part analogous to those of cardinal couples, but in part also to those of unit 
classes; for they are the smallest existent relations, just as unit classes are the 
smallest existent classes. The properties which are analogous to those of unit 
classes do not demand that the two terms of the couple should be distinct, 
i.e. they hold for i'x f i'x as well as for i'x j" i'y (where #=|=y); on the other 
hand, the properties which are analogous to those of cardinal couples do in 
general demand that the two terms of the ordinal couple should be distinct. 

The notation i l x \ i c y is cumbrous, and does not readily enable us to 
exhibit the couple as a descriptive function of x for the argument y, or vice 
versa. We therefore introduce a new symbol, "x^y," for the couple. In a 
couple x |, y, we shall call x the referent of the couple, and y the relatum. In 
virtue of the definitions in #38, this gives rise to two relations x j, and \,y; 
hence we obtain the notations x 1 "/3, i y"a, a l y, a I "/3 and so on, which 

will be much used in the sequel. It should be observed that x \, "# means 

the class of ordinal couples in which x is referent and a member of # is relatum, 

while 4 2/" a or a I V denotes the class of couples having y as relatum and a 
f> 

member of a as referent; a l "/9 denotes all such classes of couples as I y i{ a, 

where y is any member of /3; and in virtue of #40*7, s' a I "/? denotes all 

ordinal couples of which the referent is a member of a, while the relatum is 
a member of /3. This is a very important class, which will be used to define 
the product of two cardinal numbers; for it is evident that the number of 
members of s'a ], "/3 is the product of the number of members of a and the 

number of members of /?. 

The first few propositions of the present number are immediate consequences 
of the definition of x I y and the notations introduced in #38. We then pro- 
ceed to various elementary properties of the relation x 4 y, of which the most 
used are the following: 

#55"13. V: z{x] f y)w. = .z=x.w = y 

#5515. I- . D'(x ly) = i'x . d'{x I y) — i'y . 0\x ly) = i l x v i'y 

#5516. h : D<£= i'x . d'R^i'y . = .B = xly 

#55*202. Yix^y — zlw. = .x — z.y = w.=^.yix = w] f z 



SECTION Aj ORDINAL COUPLES 367 

This proposition should be contrasted with #54-22, as giving one reason 
why ordinal couples are more useful in arithmetic than cardinal couples. In 
virtue of the above proposition, when two ordinal couples are identical, their 
referents are identical, and their relata are identical. 

We proceed next to various properties of the relations x ^ and ^ x - These 
relations play a great part in arithmetic. It will be observed that if two terms 
have the relation x \, the referent is a couple whose relatum is the relatum 
in the relation x I, i.e. when we have R (x \) y, we have R = x I y (cf. #55*122). 
Similar remarks apply to the relation ^ x. The class 4 x * ' a > consisting of all 
couples whose referent is a member of a, while the relatum is x, is important. 
We have 

#55232. h : a ! 4, x"a n | y"/3 . = .# = y.a!ar»/3 
This proposition is frequently useful. 

We proceed next (#553 — '51) to give various properties of x\y which 
are analogous to the properties of unit classes. Among the more important 
of these properties are the following: 

#55-3. \-:xRy. = .xlydR.^.^l(xly)nR 
This is the analogue of #51*31. 

#55-34. h:^lR.RClxXy. = .R = xly 
This is the analogue of #5 14. 

#55*5. h:. RGxlywzlw. = z 

R = h.v.R = x\ f y.v.R = z^w.v.R = x\ f yKtz\,w 
This is the analogue of #544. 

We then proceed to such properties of ordinal couples as are not analogous 
to those of unit classes. For connecting the cardinal number 2 with the 
ordinal number 2 r , we have the proposition 

#55-54. h :: x =f= y . D :. C'R = i'x w i'y . R n R = A . = : R = x J, y . v . R =y I x 

This proposition shows that the only asymmetrical relations which have a 
given cardinal couple i'x u i l y for their field are the two corresponding ordinal 
couples x \ y and y \ x. We have next a set of propositions on the relative 
products of couples and other relations, i.e. on R\{x\ y), (x \. y) | S, and 
-^ I ( x I V) I & These propositions are very useful in arithmetic. The chief of 
them is 

#55-61. r : E ! R'z . E ! S'w . D . (R \\ S)'(* I w) = (R'z) I (S'w) 

Finally we have four propositions which belong, by their subject, to #43, 
but could not be given there, because the proofs make use of ordinal couples. 



368 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#55*01. x^y^i'xfi'y Df 

#55 02. R'x I y = R'(x I y) Df 

This definition serves merely for the avoidance of brackets. 



[(♦55-01)] 
[#38*11. #551] 
[#55-11. #14-21] 

[#55-11] 
[#55-11] 



#55-1. h . x I y = (i*x) | (t'y) 
#55*11. I- . x I l y = I y l x = x \,y=i l x'\ i'y 
#5512. h.Elxl'y 
#55-121. l-.E!4y'fl? 
#55122. h:R(xl)y. = .R = xiy 
#55123. b:R(ly)x. = .R = xly 
#55*13. V x z{x \ f y)w .= .z — x ,w = y 
Dem. 

I- . #35*103 . #55-1 . 3 h : z (x I y) w . = . z e i l x . w e i l y . 

[*51'15] =.z = x. w = y : D h . Prop 

#55*132. Y.x{x\,y)y [*55\L3] 

#55134. b.Rl(xXy) [#55-132] 

#5514. \-.xly = Cnv'ylx [*55-13.*31-131] 

#55*15. I- . D'# 4 y = i'x . d l x iy = i'y- C'x 4 y — i'x u i'y 
[#35*85*86. #51 *161] 

#5516. V : IYR = i l x . d'R = i'y. = .R = xly 

Dem. 
h. #3313*131. #51*15. D 

h :: D'R = i'x . d l R = i'y . = :. (gw) . zRw .= z .z = x: (32) . zRw .= w .w = y:. 
[#14*122] = :. (gs, w) . zRw : (gw) . zRw .D z .z = xx 

(gw, z) . zRw : (g\z) . s-Kw . D w . w = y :. 



[*11*23.*4*71] = 
[#10*23] = 

[#11*391] = 
[#14*123] = 
[#55-13] = 

[#21-43] = 



:. (g.z, w) . zRw : (gw) . zRw ,D z .z=xx (gs) . zRw.D w .w= y:. 
:. (gs, w) . zRw : zRw . D z>w . z = a; : ^i2w .D z>w .w = y:. 
:. (gz, w) . zRw : -eifo* . D Z)M , . z = x . w — y :. 
:. zRw . = Z(W ,z = x.w = yx. 
:. zRw . = ZtW .z{x\,y)wx. 
:.R = xly::2b.Prop 
The above proposition is important, and will be frequently used. 

#55-161. \-.xly=*i'll(D t R=i , a:.<I'R=*i f -y) 

Dem. 

h . #55-16 . #20*15 . D 

h . R (D'R = i'x . d'R = I'y) = R (R = x I y) 

[#5i-ii] -*'(*4y) (i) 

I- . (1) . #51-51 . P I- . Prop 



SECTION a] ordinal couples 369 

#5517. r- • x I y = ^(DH'x na'i'y) [#55'161 . *33'6'61] 
#55*2. \-:x^y — x\ f z. = .y = z 

Dem. 

\- . #30*37 . #551112 .D\-:y = z.D.xly = xl2 (1) 

h . #30*37 . #33121 . D 

bzxly = xlz.D. G'# 4 y — (I'x 4 z . 

[#55-15] D . t'y = <'* . 

[#51-23] 0.y = z (2) 

h.(l).(2).DKProp 
#55201. H:ic^^=2/^5. = ..« = y/ 
#55*202. I- : .t 4 y = .* 4 w . = . .r = £ . ;y = ?« . = . y 4 # = w 4 * 

Dew. 

h . #55-2-201 . D 

\- m . x-=z.y=w.0.x^y = z^y.z\y = z^w. 

[#13-17] D.ar4y = *4w (1) 

H. #30-37. #3312121. 3 

\- 1 x\,y = z lw .1 .T>'x ly = V>'z \,w .<±'x I y = d'z ±w . 

[#55-15] D ,i i x = i'z. i ( y = i f w. 

[#51-23] D.x = z.y = w (2) 

h.(l).(2).D 

V:x^y = z^w.~.j; = z.y—w (3) 

Similarly 

I" : ?/ 4 x = w 4 ^ • = • x ~ z • V — w W 

F.(3).(4).DK Prop 

The above proposition is important. 

#55-21. r . <P# 4 = V . d' 4 x = V [#33-432 . #55'12121^ 

#55-22. h.D'a?4=£{(ay) i2 = «4y} [*55'122] 

#55-221. h.D'4"a- = .R{(ay).jR = y4a-} [*55"123] 

#55222. h : i2 e D^4 . = . D'# =Va> . d'R € 1 

I- . #55-2216 . D h :. R e D'x 4 . ■= : (gy) . D'R = t'# . G.'R = t'y : 
[#10-35] ' = : WR = i'x : {%y) . <I'R = t<// : 

[#521] = : D'R = i'a? . d'R e 1 :. D h . Prop 

#55223. H : 72 e D< 4 x . = . (T:K = i<*' . D^ e 1 [Proof as in #55222] 
#55-224. h.D'xlnD' ly=i'(xly) 
Bent. 
\- . #55-222-223 .3 

h : i2 e D'# 4 n D'4'y . = . D'# = t'# . (I'R e I . (!<£ = i'y . V'R e 1 . 
K&w I 24 



370 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



[*52-22.*4-71] = . D'R = i'x . d'R = i*y. 

[#5516] = .R = x\,y. 

[#5115] = . R e i'{ac X y) : D I- . Prop 

#5523. \-.xl"a=R{(&y).yea.R = xly} [#38-13] 
#55 231. > . I x"a = R {(ay) • y * « • R = y 4 *} [*38-131] 
*55232. H : g ! j x"a n 4 y"$ . = .#*=?/■ 3 ! « n £ 

Bern. 
K #55-231. #1 155. D 

H :. g ! \ x ft a r\ 4 y"fi . = : (qR) : (g.z, w) . z e a . R = z 4 x . w e /3 . R = w 4 y : 
[#13'195] = : {$z, w).zea.wefi.zlx = wly: 

[#55 - 202] = : (32, w).zeoi.we^.x = y.z = w. 

[#13195] =: (a*), z eon £ . x = y: 

[#10-35] =:Rlar\0.x = y:. D H . Prop 

#55 233. r : x + y . 3 4 a"a n 4 y"£ = A [#55232 . Transp] 

The above two propositions are frequently useful in arithmetic. 



*5524. 


1- . s'x I "a = i l x f a 






Bern. 


h . *4M1 •. D 








1- :.#(«'# 4 "a)w .= 


.(aJ2).JRea?4"a.*22«/. 






[*55'23] = 


(g.fl,y).y€a..R««4y. 


zRw . 




[*13'195] = 


(3y). y ««■*(* 4 y)-w. 






[#55-13] = 


(Ry).yea.z = x.w = y. 






[#13-195] = 


z = x . wea. 






[*51-15.#35-103] = 


z (i'x f a) w :. D h . Prop 




*55241. 


h . s'4 #"« = a f t'# 


[Proof as in #5 5 -24] 





(1) 



*55'25. h : a ! a ■ => ■ D"# J, "a = i l Ox 

Dem. 

K #37-67. *33'12. #55-12.3 

h : yS e T>"x 4 "a . = . (32/) - 2/ e a • £ = D'# 4 2/ • 

[#55-15] =.(a2/).2/ea./3 = i'#. 

[*10-35] = . a '• « • £ = i'a 

h . (1) . D h :. Hp . D : £ e D'%4 "a . = . /3 = t'# . 

[#51-15] = ./3etV#:.Dh.Prop 

#55 251. h : a I a • P • d" 4 x"a = t Va [Proof as in #55-25] 

This proposition is used in the theory of cardinal multiplication (#11 31 42). 
#5526. h . d"x 4 "a = /"a [#5515 . *37'35] 

#55-261. H . D" 4 x"a = i"a [#5515 . *37'35] 

#55-262. h : 4 x"a = 4 y"/3 . D . a = £ [*55'261 . #53-22] 



SECTION a] ordinal couples 371 

#55*27. h.C"laJ tt a = C"a;l"a = J3{(sy).y€Ci./3*i t x\Ji*y} [#55*15] 
#55*28. r- :Q.'x I y = d'x \z . = .y = z . = ,xly = x\,z 
[#55*15. #51*23. #55*2] 

#55*281. H : D'y lx = D'z \ f x. = .y = z. = .y\ f x = z] f x 

#55*282. \-:C'xly = C'xlz. = .y=z. = .xly = xlz 

[#55*15*2. #54*21] 
#55*283. \-:C'ylx=C'zlx. = .y = z. = .ylx = zlx 
#5529. I- . a j (at I) = i [#55*15 . #34*42] 
#55 291. h . D | (4 a?) = e [#55*15 . #34*42] 
#55 292. h . j (x i) = 6' | (J, a*) = a£ (a = i'x o t'y) [#55*15 . #34*41] 

The following propositions, down to #55*51 inclusive, give properties of 
ordinal couples which are analogous to the properties of unit classes. 

#55 3. 1- : xRy . = .xlydli. = .'gil(x] r y)nR [#13*21*22 . #55*13] 

The first half of this proposition is the analogue of #51*2; like that 
proposition, it gives a means of reducing propositions to the form. of inclusions. 
For the second half, compare #51*31. 

#55*31. l-:xly = zliv. = .z(xly)w. = .x(zlw)y. = .x = z.y = w 
This proposition is the analogue of #51*23. 
Dem. 

b . #55*16 .Dbzxly = zlw.~. D'x ly = t'z . (I'x ly=i'w. 

[#55*15] = . i'x = i'z . t'y = t'tv . 

[#51*23] = .x = z.y = w. (1) 

[#55*13] = . x (z I iv) y . (2) 

[(1).*I3*16] ~.z = x.w = y. 

[#55*13] =.z(xly)w (3) 

K(l).(2).(3).DKProp 

#55*32. b:.xlyf\z^w = A. = : x^z.v.y^w 
Dem. 

' 1- . #55*3 . D I- : g! x | y n z \, w . = . x (z I w) y . 
[#55*13] =.x = z.y = w (1) 

h.(l).Transp.Dh.Prop 
#55*33. \-:xRy. = .xlynR = xly [#55*3 . #23*621] 
#55*34. \-:^lR.RQxiy. = .R = xly 

Dem. 
h . #55*13 . D h :. ftl R.RGx ly .=: (j-js, w) . zRw : zRw . D z>w .z = x.w = y: 
[#14*123] =xzRiv.= ttUl .z = x.v) = y\ 

[*5513] =:zRw.= z>w .z(xly)w:.Dh.?rov 

24—2 



372 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#55 341. f- :. RQx I y . = : R = A . v . R = x J, y 
Dem. 



K*4*42.DI-:.EGa-4y.= 

[*25-54] 

[#55*34] 

[#25*12] 



:RGxly.R = k.v.RGxly.R$A: 
:Rdxly.R = k.v.RQxly.±lR: 
: R G x I y . R = A . v . R = x I y : 
: R = A . v . R= x I y :. D I- . Prop 
#55 35. h:72n ; r4, ? / = A.i2c/a; > l r 3/ = ^. = .^y. J R = /Sf-^a;4,y 

h . #25*47 . D 

H:22n#,J,y==A.J?vy#4,y — S. = .xlyGS.R = S~ x\y . 

[*55"3] = .#% . R = 8 ± x\,y :D f- . Prop 

#55*36. b : xRy . = .{R^-x\,y)Kisc\,y = R 

Dem. 

h . #55*3 . D h : a-ity . = . x I y G R . 

[#23*62] =.xlyuR = R. 

[*23-91] =.(R^xly)vxly = R:2V.¥rop 

#55 37. \-:xea.ye/3. = .xly(ia'lfi 

Dem. 

h . *35103 . D I- : icea . t/ e/3 . = . «(a | /3)2/ . 

[#55*3] s.«iyG«t/8:3l". Prop 

The following proposition is the analogue of #51*232. 
#55 "4. h:.a{aj < l r i/c;24?0}&. = :a = aj.&==;y.v.a = .3:.& = w 

[#55*13 . #23*34] 
#55*41. h a R^= as I y vt z I w.D:. aRb . D a ,& . <-& (a, 6) : = . <f> (x, y) . <f> {z, w) 

Dem. 
h . #55*4 . D h ::. Hp . D :: aRb . D a , b . <f> (a, b) : = :. 

a — x . b — y . v . a = z . b = w : D a>& . <■& (a, 6) :. 
[#4*77] = :.(a,b):.a = x ,b = y .D ,<f>(a,b):a = z .b = w .D ,<f>(a,b):. 
[#11*31] = :.(a, b):a = x.b = y . D . $(a, 6):. (a, 6) : a = 2 . & = w . D .<f>(a, b):. 
[#13*21] = :. <f> (x, y) . (f> (z, w) ::. D h . Prop 

The above proposition is the analogue of #51*234. The following pro- 
position (#55*42) is the analogue of #51*235. 

#55*42. V ::R = x] f yvz\,w. D :.(g;a, 6) . aRb . <f>(a,b) . = : <f>(x,y).v . <j>(z,w) 

Dem. 
\- . #55*4 . D I- ::. Hp . D :: (ga,&) . aRb . 0(a, 6) . = :. 

(ga, 6) :. a = x . b = y . v . a = s . 6 = w : <-& (a, 6) :. 



[*4*4] = 
[#11*41] = 
[#13*22] = 



• (*RCf>, b) : a = x . b = y . <f> (a, b) : v : a = z . b = w . <f> (a, b) :. 

• (a tt » b) . a = x . b = y . (a, 6) . v . (ga, b) . a = z . b = w . <f> (a,b) :. 
.<f>(x,y).v.<f>(z,iv)::.D\-. Prop 



SECTION a] ordinal couples 373 

#65*43. \-:a:lywzlw = a:lywcld. = .z='C.w = d. = .zXw = cld 

This proposition is the analogue of #5 1*41. 

Dem. 
H . #55*202 .Dhiz = c.w = d.y.ziw = cld. 

[#23*551] D.a;lyvzlw = xlyvcld (1) 

h . #23*58 . Dt-:.xlyK/zlw = ieXywcld.D: 

z I w G x I y w c I d . c I d G x I y o z ^ w : 
[#55*3*13.*23*34] D :z=x . w=y . v .z = c ,w = d: c=x .d = y.v.c=z.d = w: 
[#13*16] D :z=x . w—y . v ,z = c ,w = d: c=x .d = y . v .z=c.w=d: 

[#4*41] D :z=x.w—y ,c=x .d = y . v.z = c.w = d: 

[#13*172] D:z=c.w=d (2) 

\-.(l).(2).Dt-:xlywzlw = xlyvicld. = .z = c.w = d. (3) 

K(3).*55-202.Dh.Prop 

#55*431. H :. a; ^ y «* « 4 w = « i & w c X d . D : 

Dem. 
h . #554 . D h :: Hp . = :.M = #.t> = y.v.w = ,s;.i> = «;: 

=u,»: w = a.» = 6 . v . u = c. v = d :. 
[#11*1] D:.x = x.y — y.yf.x = z.y — w: 

= '.x = a.y — b.^.x = c.y — d:. 
[#13*15] 5:.x = a.y = b.v.x = c.y = d (1) 

(- . #55*43 .D\- m ..x=a.y = b.D:xlywzlw = albwzlw: 
[#13*171] "D :Uip.D .albvt z lw = albv c Id . 

[#55'43] D.z = c.w = d (2) 

H.(2).Comm.#4'7.DH:.Hp.D:a;=«a.y = 6 . D .x = a.y=b.z = c.w = d (3) 
Similarly h:.Hp.D: a? = c .y — d. D ,x = c .y — d.z = a . w — b (4) 

H . (1) . (3) . (4) . D r . Prop 

*55*44. h:.a?^yc/^4«/ = a4 r ^ i:,c 4^- 

= :x = a.y = b.z = c.w=d.v.x = c.y = d.z = a.w = b: 

= :a;4y=a4 6.^4 W==C 4^' V - ^ 4 2/ = c 4 d 'Z ] e w = a \,b 
Dem. 

h . *55'43 . 7>h:x = a.y = b.D.xXy\yz^w = a\bwzlw: 

z = c.iv = d.D.albwzlw = albwcld: 

[#3*47 .*13'17] "5b:x=a.y = b.z = c.w = d. 

D.wlywz^w = albvcXd (1) 

Similarly \-;x — c-y = d.z = a.w = b. 

D.xlywzltv = alb\ycld (2) 
I- . (1) . (2) . #55-431-202 . D \- . Prop 

The above proposition is the analogue of #51 "43 



374 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*55-5. \-:.Rdxlywzlw. 

= : R=A.v.R = ply.v.R = zlw.v.R = %lyvzlw 
Dem. 

h.*2512.*2358-42.D 

h :*R = A.v .R = xly . v .R = z \,w . v .R = a >lyvz \,w: 

D .RQ.x lyvz lw (1) 

K*2549. 3\-:.R<l%ly\Jzlw.Rna:ly = A.D:R<lzlw: 

[*55'341] 3:R = A.v.R = zltv (2) 

K*25'43. D\-:.RGxlyvzlw.3:R-xlyGzlw: 

[*55-341] 3:U-xly = A.v.R^-wly=zlw: 

[*25'24.*23-551] D :(R -xly)vxly = x ly .v . 

(R-xly)vxly = xlyvzlw (3) 

r .*D5-3-36.D\-:Rl(Rna:ly).D.(R -xly)vxly = R (4) 

h.(3).(4). D\-:.RGwlyvzlw.'&l(Rnwly).y: 

R = x\ly. v . R = xLy vjzI w (5) 
h.(2).(5). -Dhz.RGxlyvzlw.y: 

R = A.v.R—x\,y.v.R = z^w.v.R = xlywz] f w (6) 

K(l).(6). Dh.Prop 

The above proposition is the analogue of #544. 
*5551. \-:.RGxlyx>S.D:xRy.v.RGS 

Dem. 

K*55'3. 3\-:Kl(Rnxly).y.xRy (1) 

K*25-49. 3\-:Hy.^^l(Rrsxiy).D.RdS (2) 

h.(l).(2).DKProp 
In the remainder of the present number, we are concerned with properties 
of ordinal couples which have no analogues for unit classes. 

*55 52. h . (i'x v i l y) | (i'z u i l w) = x Izvx ^wwy ^zvy ' \,w [#35'82-413] 

*55-521. h:x$y. = .xlyGJ [*55*3 . *5011] 

*55 53. h :. x^y .O : G l R=i i x u i'y .RQ.J . = . g ! R.RGxlyvy \, x 

Dem. 
h .*55-5.'2\-i.'&lR.RG.xlyvylx. = : 

R = x ^y .v . R = y Ix.v .R = x lyvy lx (1) 
h . *5515 . D K . C# 1 1/ = i'x u t'y ,.(7'y J, x = i f x u t'y (2) 

h . (2) . *33'262 . D h . C"(# 4, y c; y \, x) = i'x w i'y (3) 

f-.*55*521. Dh-.x^y.D.xiyQJ.yixQJ. (4) 

[*2359] D.^lyuylaGJ (5) 

h.(l).(2).(3).(4).(5).Dh:. 

x^y.D-.RlR.RGxlyvylx.'D. C'R = i'x vi'y.RGJ (6) 
h . *35'91 .Dhz&R^i'x.sji'y.O.RG (i'x u i'y) f ( t '« u i< y) . 
[#55'52] 3 . R G # 4 x ° * 4 2/ w 2/ 4 # ° y 1 2/ CO 



SECTION a] ordinal couples 375 

I- . #50-24 . D h : RXZ J. D . ~ (xRx) . ~ (yRy) . 

[*55'3.Transp] D . i2n*|«= A . R nyl y = A (8) 

t- . (7) . (8) . #2549 . D h : C'R = i'x w t't/ . R G J". D . R <Zx I yv y I x (9) 

I- . *33-24.*5M61 . D I- : C'R = i l x u i*y . D . g ! R (10) 

h . (9) . (10) . D h : <7<i2 = i l x yj t'y . R G J . D . a ! i2 . R G J, 3/ c; y \,x (11) 
I- . (6) . (11) .31-. Prop 

#5554. \-::x$y.D:.C'R=i'xvi'y.Rr\R = A. = :R = xly.v.R = yXx 

Dem. 
K #50 -46. #471 . D\-:RnR = A. = .RGJ.RnR = A (1) 

I- . (1) . #5553 . D I- :: as + 2/ • 3 :■ C"# = i^ w t'y . E A R = A . 

EEia'.iZ.jKGajlyc/yJptf-l^i^A: 
[*55'5'134] =:R = xly.v.R = ylx.v.R = xlyvylx:RnR = A (2) 
I- . #55 - 32 .0\-:.x^y.Dzxly.oylx = A: 
[*55'14] D:R = xly.3.RfxR = A: 

R = ylx.D.RnR = A (3) 

h . #5514 . #3115-33 .2\- :R = x lyvylx .D . R = R. 
[#23"5] 5.Rnil = R. 

[♦55-134] 3.<&lR*R (4) 

K(3).(4).*4-7l.*5-7l.D 

= \R = x\,y . v.R = y ^x (5) 
h . (2) . (5) . D r . Prop 

♦5557. V.R\(x^y) = R'x^i'y [♦3781 . #551 . #53301] 

♦55571. r . (a? | y) \ S = i'0 f S 7 ^ 

♦55-572. r- . ^K^ | y)|fl«"5'a? f Sty [#55-571 . #37-81] 

♦55-573. h . ii|(0 4 y)\S=R'x \~S'y [#55-572 |~| 

#55-58. H:E!E £ 0.D.ie|(0 42/) = ( i2 ^)4y [#55-57 . #53-31 . #55-1] 

#55-581. h:E!S t t/.D.(0 4,3/)|^ = 4(S'y) 

#55-582. I- : E ! R l x . E ! S'y . D . R j (x I y) \ S = (R'x) X (S'y) [#55-58-581] 

#55-583. h:El R'x. El S'y. D.R\(xly)\S = (R'x)l(S'y) [~*55-582 ^1 

The above propositions are frequently useful in arithmetic. Their use 
arises as follows. Let a, ft 7, B be classes of which a is correlated with 7 by 
the relation R, and £ with 8 by the relation S. Then if x e 7 . y e B, the 



376 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

couple consisting of the correlate of x and the correlate of y is (R'x) \ (S'y), 

i.e., by the above, R | (x J, y) \ 8, i.e. (R\\S)<(xly). Thus the relation R\\8 
correlates the couples, in a and /3, composed of the correlates of terms in 
7 and 8. The most useful form, in practice, of #55583, is that given below 
in *55'61. 

*556. h.(R\\S)'(zlw) = R'zfs<w [#55-573 .#43-112] 

#55-61. h : E ! R'z . E ! &'w . D . ( R \\ S)<(z I w) = (R'z) j (S'w) 
[#55-583. #43-1 12] 

#55-62. \-:2$iv.8 = xlzvylw.D. S'z = x.S'w = y 
Dem. 
f- . #5513 . D h :: Hp . D :. u8z . = : u — x . z = z . v : . u = y . z = w (1) 

h . (1) . #13-15 . D V :. Hp . D : uSz . = . u = x (2) 

Similarly I- :. Hp . D : uSw .=*u = y (3) 

r . (2) . (3) . #30-3 . D r- . Prop 

#55-621. I- :x^y.S = xlzvylw.D.S'x = z.S'y = w 
[Proof as in #55*62] 

The four following propositions belong to #43, but are inserted here because 
the proof uses #55- 13. 

#55-63. r :g! Q *S . P\\Q = R\\S .D . P = R 
Bern. 
\-.*4,3'm.D\-::Uv.D:.P\(ylz)\Q = R\(ylz)\S:. 
[*34-l] D :. (gw, v) . xPu .u(y\z)v. vQw . = XiW . 

(aw, v) . xRu -u(y ],z)v . vSw :. 
[*5513.*13*22] D :. xPy . zQw . = x>w . xRy . zSw :. 

[#4-73] D:.zQw.z8w.0 w :xPy.= x .xRy (1) 

h . (1) . #1011 . #11-35 . D h :. Hp . D : xPy . = x . #% (2) 

K (2). #1 011-21. Dh. Prop 

#55-631. \-:&lPnR.P\\Q = R\\S.D.Q = S [Proof as in #55 63] 
#55-632. h : P\\Q = R\\S .Rl P .Rl Q .3 .Rl P * R . Rl Q n S 

Dem. 

K*5513. D\-:xPy.zQw.D.x{P\(ylz)\Q}w. 

[#43-112] D.x{(P\\Q)<(ylz)}w (1) 

h .(1) .Dh :.R V .D :xPy . zQw .D . x {(R || £)<(?, j s)j w . 
[#43-112] D.*{i2|(y4*)ii8}«;. 

[#34-1] D . (gw, t>) . xRu ,u(ylz)v. vSw . 

[*55-13.*13-22] D . xRy . *#w . 

[*4-7] D.x(PnR)y.z(QnS)w:.Dh. Prop 

#5564. H:.a!P.a!Q.v.a!i*.a!£:D:P||Q = i2||#. ==.P = R.Q = S 
[#55-63-631 -632] 



*56. THE ORDINAL NUMBER 2,.. 

Summary of #56. 

In this 'number, we have to consider the class of those relations which are 
each constituted by a single couple. In case the two members of this couple 
are not identical, the class of such relations is (as will be shown later) the 
ordinal number 2, which, to distinguish it from the cardinal number 2, we 
denote by "2 r ." (Here the suffix is intended to suggest "relational") The 
class of all relations consisting of a single couple, without the restriction that 
the two members of the couple are to be distinct, will be denoted by "2\" 
This is not an ordinal number. It will be observed that there is no ordinal 
number 1, because ordinal numbers apply to series, and series must have 
more than one member if they have any members. This will appear more 
fully when we come to deal with series. 

The properties of 2 are largely analogous to those of 1, while the properties 
of 2 r are more analogous to those of 2. 

Most of the propositions of the present number are seldom referred to 
in the sequel, but such references as occur are important. The most useful 
propositions in the present number are the following: 

*56111. \-:Re2 r . = .D<R > <I'Rel.I>'Rr>a'R = A 

#56112. \-:Re2 r . = . D'ftd'Rel .C'Re2 

#56113. K2 r =2n5"2 

Observe that "V"2" means "relations whose fields have two terms." 
*5613. b.2-2 r = R {( a a) . R = a I a} 

*56'37. r:i2e2 r . = . C'Re2.RnR = A 

I.e. 2 r is the class of asymmetrical relations whose fields have two terms. 
#56381. \-:C'R=i'x. = .R = xlx 

#5639. K2-2 r =C"l 

I.e. the relations which are couples whose referent and relatum are 
identical are the relations whose fields consist of a single term. 

#5601. 2 = R {fax, y) . R *» x | y\ Df 

*56'02. 2 r = R{(Rx, y).x$y.R = xly] Df 
*5603. r =*'A Df 

#661. \-:Re2. = .(Rx, y) . R = x± y [*20'3 . (#5601)] 



378 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



*56 101. h : R e 2 . = . D'R, d'R e 1 

Bern. 

f .*5516.*11 11-341. D 
•""■•(a^ y).R = x\,y % = 
[*ll-54] = 

[#52-1] = 

V . (1) . #56-1 . D V . Prop 



(gar, y) . D<£ = t'a? . d'R = t'y : 
(gar) . D'R = t'ar : (gy) . (I<12 = t'y : 

D'iz, a-j? 6 i 



(i) 



*56102. h.2 = D"ln<I"l 

Bern, 

h . *56-101 . *37106 . D 

\-:Re2. = .ReD"l.R€d"l. 

02233] =.JKeD"ln5"l:DI-.Prop 

*56103. l-:12e2.D.g!l2 

Dem. 

h . *56101 . D H : 12 e 2 . D . D<12 e 1 . 
[#5216] D.g!D<12. 

[*33-24] D . 3 ! R : D h . Prop 

*56104. \-:R € r . = .R = A [(*5603)] 

«66'11. H : 12 e 2 r . = . (ftw, y).a- + y.l2:=a4y [*20'3 . (*5602)] 

*56111. h : R e 2 r . = . D'12, CF12 e 1 . D'R rx d'R = A 
Dem. 
K*51-231.*55\L6.D 

I- : x '+ y . 12 = x | y . = . t'x r\ t'y = A . D'12 = t'a; . d'R = i'y . 
[*13*193] = . D'R n d'R = A . D<12 = t'x . d'R = t'y (1) 

h.(l).*5611.*ll-ll-341.3 

I 1 :. 12 e 2 r . = : (gar, y) . D<12 n (F12 « A . D'R = t'a; . <P12 = t'y : 
01145] = : D'R n d'R = A : (rx, y) . D'12 = i'x . d'R = t'y : 
011-54] = : D<12 n d'12 = A : (gar) . D'12 = t'x : (gy) . d'12 = t'y : 
[*521] = : D'R a d'12 = A . D'R, d'R e 1 :. D h . Prop 

*56 112. I- : R e 2 r . = . D'12, d'R el.C'Re2 

Bern. 

h.*56111.*54-43.3 

r- : R e 2 r . = . D'R, d'R el . D'12 v d'12 e 2 . 

03316] = . D'12, d'R e 1 . C'R e 2 : D t- . Prop 

*56113. K2 r = 2nC"2 

Dem. 

H . *56112K)1 . D f- : R e 2 r . = . R e 2 . C'R e 2 . 

[«37*106.«33-122] =.ReZ.ReC"2. 

022-33] . = . R e 2 n C"2 : D I- . Prop 



SECTION A] THE ORDINAL NUMBER 2 r 379 

*56114. h.2 r = £"ln<I"lnO"2 [*56-113102] 
*56'12. \-:Re2 r . = .R € 2.RQJ 
Dem. 

h.*55-3.*5011. >Y:x^y. = .xiyQ.J: 

[Fact] "5\-:R = xly.x^y f = .R = xly.xlyGJ. 

[*13193] = .R = x\y.R<iJ (1) 

K(l).*lll 1-341. D 

I- :. (a#, y) . R = x I y . x^y . = : (a#, y) . R = x I y . R G J: 

[*1145] =i{^x,y).R = x\ f yiRdJ: 

[♦56-1] = :Re2.RGJ (2) 

h . (2) . *5611 . D t- . Prop 
*56121. K2 r Gi [*56113] 

*56122. H: J Re2 r .D.a!i2 [*56121103] 

*5613. h. 2- 2 r = R{(<&a).R = a 4 a} 

I- . *5611 . *1152 . Transp . D 

H:.R~e2 r . = :R=sx^y . D Xty .x = y (1) 

f- . (1) . *561 . 3 

h : . jB e 2 — 2 r . = : (ga, 6) . i2 = a ^ & : R = « 4 y - ^*,» • #= y s 

[#11 45] = : (ga, &):U = a4&:-S = #4y- 3<r,jf >x—y. 

[*13'193] =:(<^a > b):R = aib:alb = xly.D x>y .x = y: 

[*55202] = :(ga,&): R-albza = x.b = y .D Xt y.x = y: 

[*13'21] =i(fta,b).R = alb.a = b: 

[*13-195] = : (ga) . £ = a I a :. D h . Prop 

2 — 2 r might be defined as the ordinal number 1, sinee it is what we shall 
call a relation number (cf. #153). But we wish bur ordinal numbers to be 
classes of serial relations, and such relations have the property of being con- 
tained in diversity. Hence if we were to define 2 — 2 r as the ordinal number 
1, we should introduce a tiresome exception, from which trivial complications 
would be introduced into ordinal arithmetic. We have, therefore, not adopted 
this course. 

#5614. r . D f (x 4>= 2 rs Ib'i'x 
Dem. 

f- . *33-6 .31-: D'R = t f x. = .Re T>H'x (1) 

h.(l).*561.D 

h :. R e 2 n D'i'x . = : (%z,y) .R = ziy: D'R = i'x i 

[#55*16] == :(Rz,y) rT>'R=: i'z .d'R = t'y :D'R=i'x i 

[*ll-45] = : ( a *, y) . D'R = i<z . d'R = i*y . D'R = t'x : 



380 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



[*13-193] 

[*51-23] 

[*13-195] 

[*5516] 

[*55-22] 



(g«, y) . T>'R - i'z . d'R = i'y . i'z = t'x : 
(g*,y) . D'i2 = t'.s. (Pi? = i'y.z = x: 
(gy) . D<fl = i l x . (F22 = t'y : 

i?€DV|):.Dh.Prop 



«66'141. h . D'| x = 2 n d'l'ff [Proof as in *5614] 

*5615. I- . D'(* I) - i\x 4, x) = % WV'i'x 

Dem. 
b . *5522 16 . D r :. R e {D<(# |)j - t'(a? J, a?) . 
(gy) . WR = t'x . d'R 



i'y:~(D'R = i'x.<l'R=i,'x) 
(32/) . D'i2 = t'x . d'R = t'y . ~ (CI'12 = t'a?) : 
(%y) . D'i* = i l x . G'R = i l y . ~ (i'y = l'#) : 
(gy> • P'-R = i'tc . d'R = i'y . « + y : 

(a*, y) • * + y ■ D '^ = «'* • a'^ = t'y • D'.R = 1<X '• 

(a*» y)- z *y- V'R = *'* . <i'# = i<y : d^ = i<x -. 

<— 

JSe2 r nD f t^:.Dh.Prop 

h . D'(4 #) - i\x I x) = 2 r ntl'i'tf [Proof as in *56*15] 
f- .x^ye2 



[*10-35.*4-51.*5-61] = 

[*13193] = 

[*51'23] = 

[*13195.*5123] = 

[*13-193] = 

[*ll-45] = 

[*55-16.*336] = 

[*5611.*2233] = 

*56151. 

*5616. 

Dem 



*5617. 

Dem. 



*5618. b 
Dem. 



b . *21*2 .Oh .xly = xly . 
[*11'36] Db .(^z,w).xly-zl w. 
[*561] Dr.a^ye^.Dr.Prop 
xly€2 r . = .ylx€2 r . = .x^y 

K*5611.D 

b :. x ^ y e 2 r . = : (gz, w).^4 :W ' ;:c 4y = ' 2r 4 w: 

[*55"202] = : (rz, w) .z^w .x = z .y —w: 

[*13-22] =:x^y 

Similarly 

b:ylxe2 r . = .x^y 

K(l).(2).DKProp 

x~e a . = . x J, "a C 2, . = . j x"a C 2 r 

h . *13'196. D h :.a;~e« .siyea.D^.yfa?: 
[*56"17] =:yea.D y .xlye2 r 

[*37-61.*38-1211] = : x I "a C 2,. 

Similarly I- : x ~ e a . = . I x"a C 2 r 

b . (1) . (2) . D b . Prop 



(1) 

(2) 



(1) 

(2) 



SECTION A] 



THE ORDINAL NUMBER 2 r 



381 



*5619. V:Re%..xe 1 D t R. = .{'g L y).x$y.R = xiy. = .Rex\ f "-i ( x 

Bern. 
\- .*5frll .*11 4,5 .Dt- :. R € 2 r .xeD'R . =:(Ry,z).yJrZ . R = y I z .xeV'R: 
[*55'15] = :('^y,z).y^z.R = yiz.X€i t y: 

[#51-23] =:(fty,z).y$z.R = ylz.x = y: 

[#13195] = :(%z).x±z.R = xiz: (1) 

[#5115] ^-.(^.ze — l'x.R^x^z; 

[*38-13] =-.RexX tf -i ( x (2) 

K(l).(2).Dr.Prop 
#56191. h:Re2 r .xe<I'R. = .(Ry).x$y.R = ylx. = .Relx"-<, t x 

[Proof as in #5619] 
#56*2. b:.Re2.=a:('&x ) y):zRw.= ZtW .z = x.w = y [#55*13 . *56'1] 

#56.21. \-:.Re2. = :^lR:xRy.zRw.D XtViZ>w .x = z.y=:w [#56'2.#14'124] 
*5622. h . A ~e 2 [#56103 . #2553] 

*56'24. h.a!2. a !-2 [#56-22-16 . *10'24] 

*56'25. h.24=AnRel.2 + VnRel [#56-24 . #24-54-17] 
#56-26. 1- :. Ret u t'A. = :xRy . zRw . D x ,y,z,w x = z.y = w 

This proposition is the analogue of #52-4. 

Bern. 
K #51236. DK::#e2vi'A. 

= :.Re2.v.R = A:. 
[*25'51] =:.Re2.v.~±lR:. 

[*56'21] = :. a ! R : xRy . zRw . ~5 x , y ,z,v> -x = z 

[#562] = :.xRy.zRw.'D Xt y iZyW .x = z.y — w 

h . #11-36 . Transp . D h :. ~ g ! R . D : ~{xRy) . ~ (*ftw) : 
[#2'2-l] D : #JRt/ . D . a; = z : zRw .3 .y = w : 

[*3-47] "2:xRy.zRw.D.x = z.y = w (2) 

h. (2). #1111-3. Dh:.~a!i2.D:«%.^tf;.D ariyiZ(W .^ = ^.y = w (3) 
h . (1) . (3) . #4-72 . DKProp 

#56261. h::Re2.D:.SGR. = :S = A.v.S = R 

Bern. 

h. #55-341.3 I- ::£=# J, yD:. SGR . = : S = A.v . S=R (1) 
h . (1) . #1111:35 . #561 . D K Prop 

#56262. \-:.Re2.-D:SGR.<&lS. = .S = R 

Bern. 

\-.*56-22.Db:.Re2.D:S=R.3.S$A (1) 

l-.(l). #5-75. #56-261. D 

h:.Re2.3:SGR.S^A. = .S=R (2) 

I- . (2) . #2554 . D h . Prop 



y=w:. vz.^qIR:. 
v.^rIR (1) 



382 PROLEGOMENA TO CARDINAL ARITHMETIC [EART II 

#5627. h:.Re2.y-.<gilRnS. = >RnSe2 

Dem. 

h . #5534 . #2343 . D 

\-:.B = xly.O:<&lRrxS. =.RnS=R. 

[#56-16] D.RnSe2 (1) 

h. #56-103. Dh:RnSe2. D.RlRnS (2) 

l-.(l).(2). DI-:.^ = A- > |,y.D:a!i2n. < ?. = .i2n/Se2 (3) 

h. (3). #1 111-35. #561. >r. Prop 

*56'28. \-:.Re2.D:RlRnS. = .RGS. = .RfsS = R 

Dem. 

\-.*b5-3.1\-:.R = xly.D:<zlRnS. = .R<iS. (1) 

[*23'621] =.RnS = R (2) 

h . (1) . (2) . #1111-35 . #561 . D K . Prop 

#56281. \-:.Re2 r .D:'&lRnS. = .R<iS.~.RnS = R* = .RnS€2 r 
Dem. 

I- . *56121 . D I- :. Hp . D : JR e 2 : 

[#56-28] D:^lRnS. = .RQS. = .RnS^R (1) 

K #1313 .31-:. Hp . 0:RnS = R.D.Rn8e2 r : 

[(1)] D:g!linS.D.JJnSe2 r (2) 

»-. #66-122. DH:JBA/8fe2 r .D-a!Bnflf (3) 

K(2).(3).DI-:.Hp.D:a!i2nS. = .i2ASe2 r ( 4 ) 

K(l).(4).DKProp 
#56-29. h::P, Qe2.D:.PGQc/P. = :P = Q.v.PGi* 
Dem. 

K. #55-51. D 

V :.x ^yQ.z Iwv R ."5:x{z ^w)y .v .x ),yQ.Ri 

[#55*31] "D:xly = zlw.v.xly<ZR (1) 

K(l). #1312.3 

\-::.P = xly.3::Q = zlw.3:.PGQvR.0:P = Q.v.PGR (2) 

K (2). #11-11-35. #561. D 

H::.P € 2.D::Q = ^4w.D:.PGQc; J R.D:P = Q.v.PCi2 (3) 

K (3). #11-1 1-3-35. #561. 

l-::.P€2.D::Qe2.D:.PGQc; J R.D:P = Q.v.PGE (4) 

h.*23-58-61.3l-:.P = Q.v.PG72:D.PGQc;i2 (5) 

h . (4) . Imp . (5) . D H . Prop 
#56-3. H:.P,Qe^.D:PGQ.= .P = Q. = .g[!PnQ 

I- . *55-3-31 . D 

h : x I y G z J, w . = . x I y = z \ w . = . g ! (# J, y) f\ (z I w) (!) 



SECTION A] THE ORDINAL NUMBER 2 r 383 

K(1).*13'12.D 

\-:.P = xly.Q = zlw.D:P(ZQ. = ;P = Q. = .zlPnQ (2) 

r . (2) . *11 1135 . *561 . 3 h . Prop 

The steps from (2) to the conclusion are analogous to those from (2) of 
#56*29 to the conclusion of #56*29. Analogous steps in succeeding proofs 
will be merely indicated as above. 

*56'31. b:.P, Qe2.D:P=j=Q. = .PnQ = A [*56*3 . Transp] 

*56*32. h:Pe2.D.PnQ e 2wM 
Bern. 

h .*56-27 . D h :. Hp. D : g ! Pn Q. D . PnQe2 : 
[*2*54.*25*54] D : Pn Q = A. v.PnQ e 2 : 

[*51*236] 3 : P n Q e 2 u t'A :. D h . Prop 

*5633. \-::P,Qe2.D:.R(ZPuQ. = :R = A.v.R=P.v.R=Q.v.R=PuQ 

Dew. 

h . *55*5 . *1312 . D h :: P = # I y . Q = * J, w . D :. 

-RGPeiQ. = :£ = A.v.P = P.v.# = Q.v.i2 = PoQ (1) 

I- . (1) . *1111*35 . *561 . D I- . Prop 

*56*34. H::P,Q 6 2.P=f=Q.D:.JSGPc;Q.a!E. J B4=Pc/Q.=: J R=P.v.i2 = Q 

I- . *56*33 103 . *575 . *25*54 . D 

h::P, Q 6 2.D:.i2GPc;Q.a!i?. = :P = P.v.i2 = Q.v.i? = PoQ (1) 

K*23*62. Df-:P = PiyQ. = .QGP: 

[*56*3] Dh:.P, Qe2.D:P = PvQ. = .P = Q: 

[Transp] D : P+Q . D . P + Pc» Q:. 

[*13*181] Dh:.P, Qe2.P$Q.3:R = P.3.R$PvQ (2) 

f--(2)^|.3l-:.P,Q62.P=|=Q.D: J B = Q.D.i2 + Pc;Q (3) 

H.(2).(3). D\-::P,Qe2.P$Q.D:.lt = P.v.R=Q:D.R$PvQ (4) 
K(l).(4).*5*75.DKProp 

*56 35. I- : C'R e 2 . R n 5 = A . D . # e 2, 

H . *55*54 . D 

r- :. « + y . C'P = i'xv I'y . R* R = A .2 : R = xly .v . R = y lx: 
[*56*17] D:jRe2, (1) 

I- . (1) . #11-1 1-35 . *54101 . D h . Prop 



384 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#5636. \-:Re2 r .D.C'Re2.RnR = k 
Dem. 

K*55-54.D 

\-:x$y.R = xly.D.x$y.C'R = L'xvi<y.RnR = A (1) 

K(l). #111 134. #5611.:) 

I- :. Re 2 r . D : (rx, y) . x^y . C'R = i'x v t'y . R « R = A : 
[#54101. #1 145] D : C'R e 2 . R n R = A :. D I- . Prop 

The following proposition, in addition to being used in #56*38, is used in 
the elementary theory of series (#204-463). 

#5637. \-:R € 2 r . = .C'Re2.RnR = A [*563536] 

*5638. \-.2 r = C"2rsR(RnR = A) 

Dem. 

K #37106. #33122. DHC'Ee 2. s.EeC"2 (1) 

K#20'3. 3\-:RnR = A. = .ReR(Rf>R = A) (2) 

h.(l).(2).*56-37. Dh:Re2 r . = .ReC"2.ReR(RnR = A) . 

[*22'33] = . R eC"2 nli(RrxR = A): D KProp 

This proposition is important as establishing the connection between the 
cardinal and ordinal 2. It shows that the ordinal 2 consists of those asym- 
metrical relations whose fields have (cardinal) 2 terms. It is used in the 
theory of well-ordered series (#250-44). 

The following proposition, in addition to being used in #5639, is used in 
relation-arithmetic (#165 - 38) and in the theory of series (#205 - 4). 

#56381. \-:C'R = i t x. = .R^xlx 

Dem. 

I- . *3324161 . #51161 . D h : C'R = i'x . D . a ! D'R . D'R C i'x . 

[*51-4] D. D'R ^ i'x (1) 

Similarly h:C'R = i'x . D . d'R = i'x (2) 

K(l).(2).*5516. Dh:C'R=L'x.D.R = xlx (3) 

h . *55'15 . D I- : R = x I x . D . C'R = i'x (4) 

h . (3) . (4) . D h . Prop 

#56 39. V . 2 - 2 r = C"\ 

h . *56'381 .Oh C" ft e I . = . (&x) . K = a; J, a; . 
■[#56-13] =.i2e2-2, (1) 

K (I). #37-106. DK Prop 



SECTION A] THE ORDINAL NUMBER 2 r 385 

This proposition establishes the connection between 2 — 2 r and 1, showing 
that 2 — 2 r is the class of those relations whose fields consist of a single term. 
It is used in the discussion of r and 2 r and 2 — 2 r as relation-numbers 
(#153-301). 

#56*4. h:./iCi.D:a!Jye/t.= .a!(s'/i)y 
Dem. 
h . #41-11 . D h :. Hp . D : a-(s*>) y . = . (&R) .ReZ.Rep. xRy . 
[#56'1] = .(qz, w).z I W€fi.x{z^ w)y . 

[#55*13] = .(qz,w).zX we fi.z = x . w = y . 

[#13-22] =.xly € fi:.Dh.Prop 

This proposition is the analogue of #53*23. It is used in the number on 
exponentiation in relation-arithmetic (#176-19). 



e&w i 25 



SECTION B 

SUB-CLASSES, SUB-RELATIONS, AND RELATIVE TYPES 

Summary of Section B. 

In this section, we consider first the classes contained in a given class and 
the relations contained in a given relation. If a is any class, the classes con- 
tained in a are the members of /8(/3 Ca); these are also called the sub-classes 
of a, or (sometimes) the " parts " of a. In this last usage, they are called 
"proper parts" when they are not coextensive with a, this phrase being formed 
on the analogy of "proper fractions." The sub-classes of a are all the classes 
that can be formed from members of a; they are the same thing as the 
"combinations" of members of a taken any number at a time. If n is the 
number of members of o, 2 n is the number of sub-classes of a, whether n be 
finite or infinite. The number of sub-classes of a is always greater than the 
number of members of a. On account of these and other propositions, the 
class of sub-classes of a given class is an important function of the class. If 
the class is a, we denote the class of its sub-classes by "Cl'a." This is a 
descriptive function, derived from the relation "CI," defined as follows: 

Cl = £a{* = /§08Ca)} Df. 

The sub-relations of a given relation are all the relations contained in the 
given relation, i.e. all relations which imply the given relation for all possible 
arguments. That is, if P is the given relation, R is a .sub-relation of P if 
RdP. Thus denoting the class of sub-relations of P by "Rl'P," we are to 
have 

m<P = R(RGP); 
hence we take as the definition of "Rl" the following: 
m = \P{\ = R(RGP)} Df. 

Sub-relations have properties analogous to those of sub-classes, but they are 
of somewhat less importance. It should, however, be observed that when one 
series is contained in another, i.e. is obtained by selecting some of the terms 
of the other series without changing their order, then the generating relation 
of the one series is a sub-relation of the generating relation of the other series. 
(It is not the case that a sub-relation of the generating relation of a series 
must generate a contained series, for its field may fall apart into detached 
portions, or otherwise fail of being serial.) 



SECTION B] SUB-CLASSES, SUB-BELATIONS, AND RELATIVE TYPES 387 

We shall also consider in this section (#62) the relation of membership of 
a class, i.e. the relation which x has to a when x e a. This relation bears the 
same relation to "xea" as "7" bears to "x — y." Strictly speaking, we ought 
to introduce a new notation for it, putting (say) 

A=xa(xea) Df. 
But as e, unlike "=," is a letter, and capable of being conveniently used 
alone, it seems more desirable, from the point of view of avoiding unnecessary 
duplication of symbols, to put 

e — xa(xea) Df. 
Strictly speaking, this definition is faulty, since it gives two different meanings 
to "e." But practically this does not matter, since the above definition gives 

h: xea. = . xea, 
where the first e has the meaning just defined, while the second has the old 
meaning. Thus all that is really required of the above definition, namely to 
give a meaning to formulae in which e occurs without referent or relatum, is 
effected without the danger of any confusion that could lead to errors. 

The chief importance of e as a relation arises from the fact that relations 
contained in e play a very important part in arithmetic. Take, for example, 
the problem of selecting one term out of each member of a class of classes: 
in this case we require a selecting relation R which is such that whenever 
xRa, # is a member of a, i.e. such that R G e. (This condition is only part of 
the definition of a selecting relation; the complete definition is given in #80.) 

Three numbers in this section (#63, #64, #65) are devoted to the discussion 
of relative types. Given a variable x, we often want to define the relative 
types of other variables, or of ambiguous symbols, occurring in the same con- 
text; that is, we wish to express the types of these other symbols in terms of 
that of x. We use "t'x" for the type of x, "t £ a" for the type in which a is con- 
tained. Then t<fa = a u — a, t'x = i'x u — l'x = tji'x, and t l a = t ' Cl'a = Cl%'a. 
Also we introduce a notation (#65) for giving typical definiteness, relatively 
to x, to typically ambiguous symbols. This notation is very useful in cardinal 
and ordinal arithmetic, since numbers are typically ambiguous, and the failure 
to take account of this fact has led to the contradictions concerning the greatest 
cardinal and the greatest ordinal. 



25—2 



*60. THE SUB-CLASSES OF A GIVEN CLASS 

Summary of #60. 

Our definitions in this number are as follows: 
*6001. Cl = £a{/e = /9(/3Ca)} Df 

This defines the relation to a class a of the class of all its sub-classes. 
*6002. Clex = £«{*: = £09Ca.a!/3)} Df 

This defines the relation to a class a of the class of all its existent sub- 
classes, i.e. of all its sub-classes except A. This is often required, as, for 
example^ in the statement of Zermelo's axiom: "Given any class a, there is 
a relation R such that, if y8 is any existent sub-class of a, R'ft is a member 
of /3," i.e. 

"(rR) : /3 e CI ex'a , D fi . ;R</3 e /3." 

This axiom, or its equivalent the multiplicative axiom, plays (as will appear 
hereafter) an important part as the hypothesis to many propositions in 
cardinal arithmetic. 

*6003. Cls 2 = Cl'Cls Df 

A Cls 2 is a class whose members are classes. 
*6004. Cls 8 = Cl'Cls 2 Df 

A Cls 3 is a class whose members are classes whose members are classes, 
i.e. a Cls 3 is a class of classes of classes. 

Apart from propositions which merely embody the definitions, the most 
useful propositions in this number are the following: 

*60-3. KAeGFa 
*60-32. h.Cl'A=VA 
*60-34. h . a e Cl'a 
*60-362. h.Cl'i'x^t'AvL'L'x 

I.e. A and i'x are the only sub-classes of a unit class i'x, 
*60-5. h.s'Cl'a = a 
*60-57. K«CC1V* 
*60*6. \-:xea.D.i'xeC\ex'a 

The propositions of this number are chiefly useful in cardinal and ordinal 
arithmetic, but uses also occur in the theory of series; hardly any uses occur 
before cardinal arithmetic. 



SECTION B] THE SUB-CLASSES OF A GIVEN CLASS 389 

#6001. Cl = £a{/e = £(/3Ca)} Df 

#6002. Clex = £a{/e = /§08Ca.a!/3)} Df 

#6003. Cls* = Cl<Cls Df 

#6004. Cls 3 = Cl'Cls 2 Df 

#601. I- :* CI a. = .* = /§ (£ C a) [#21-3 . (#60-01)] 

#6011. h:/ C Clexa. = ./c=/3(y3Ca.a! y 8) [*2 1-3 .(#60-02)] 

#6012. KCl'a = £(/3Ca) [*30'3 . #601] 

#6013. K.Clex'a = /§09Ca.a!/3) [#30*3 . #6011] 

#6014. KElCl'a [#6012 . #14-21] 

#6015. KElClex'a [*60'13 . #14-21] 

#60-2. H:/9eCl'a. = ./3Ca [#6012 . #20-33] 

#60-21. h:/3eClex<a. = ./3Ca.a!/3 [*6013 . #20-33] 

#60-22. h:/3eClex'a. = .#eCl<a.a!y8 [*60-2'21] 

#60-23. h:/3eClex'a. = .y8eCl'a-t'A [#60-22 . #53*52] 

#60-24. KClex'a=Cl'a-t'A [#6023 . #20-43] 

#60-3. KAeCl'a [*2412 . #60-2] 

#60-31. h.glCPa [#60*3 . #10*24] 

#6032. h.Cl'A = t'A 

Bern. 

V . #60-2 . #2413 . D V : aeCl'A . = . a = A . 

[#5115] H.aet'AOh.Prop 

#60321. H:a = A. = .Cl'a = t'a 

Dem. 

H.*60-32. DI-:a = A.D.Cl'a = t<a (1) 

H.*60-2.*5115.D 

H:.Cl'a=i'a. = :£Ca.=0.£ = a: 
[#101] D:ACa. = .A = a: 

[#2412] D:A = a (2) 

K (1) . (2) . D K Prop 

#60-33. KClex'A = AnCls 

We Avrite "An Cls" on the right, to indicate that the A concerned is of 
higher type than the A on the left. 

Dem. 

h . #60-22-32 . D h : /3eCl ex'A . = . 0e l'A . g ! /3 . 
[*51-15.*24-54] = ,£ = A.£+A (1) 

K(l).*3-24.DK/3~eClex'A (2) 

I- . (2) . #1011 . *2415 . D h . Prop 



390 PROLEGOMENA TO CARDINAL ARITHMETIC [PART TI 

*60-34. KaeCl'a [*22'42 . *60'2] 

*60'35. l-:g!a.D.aeClex'a [*6022-34] 

*60-36. hrgSa.D.glClex'a [*60"35 . *10"24] 

*60361. h:a!a. = .g!Clex'a [*60-3633] 

*60362. b.Cl'i'x^i'Ayji'i'x [*51-401 . *60'2] 

*60'37. h.C\ex'i'x = i<i'x 

Dem. 

h . *60'21 . D h : /3 e CI ex'i'x . ~ . C i l x . g; ! . 

[*51-4] =.j3=i'x. 

[*51-15] = . /3 e t'i'tf : D h . Prop 

*60371. Hael.D.Cl'aCOul 

Dem. 

\- . *51'401 . D h : : a = i*x . D :. C a . = : = A . v . = i l x : 

[*54-102.*52-22] D : e . v . e 1 : . 

[*60-2.*22-34] D:.0€ Cl'a . D . e u 1 (1) 

I- . (1) . #10-1 1-23' . *52 1 . D h . Prop 

*60 38, I- : a e 1 . = . CI ex'a = t'a 
.Detw. 

h . *60-37 . Dh:a = i*w.D.C\ ex'a = i'a : 

[*1011-23] D h : (gp) . a = i'x . 3 . CI ex'a = i'a : 

[*521] DH:ael.D.Clex'a = .i'a (1) 

h . *60'361 . *51161 . D f- : CI ex'a = i'a . D . g ! a (2) 

h . *60-21 . *10'1 . D h :. CI ex'a = t'a . D : t'# C a . g ! i'x . = . t'# = a : 
[*5M61] Du'aCo. = .i'tf = o: 

[*51'2] D:#ea. = .t'#=a (3) 

t- . (3) . *10 11-21281 . Dh:.Cl ex'a = t'a . D : g! a . = . (a*) . t'ar = a . 
[*52-l] = . a e 1 : 

[(2)] D:ael (4) 

I- . (1) . (4) . D h . Prop 

*60 39. \- . Cl'(i'x u t'y) = t'A u i'i'x yj t'l'y v i'(t'# v t'y) [*54-4 . *602] 
*60'391. h:ae2.D.Cl'aC0v,lu2 [*54-411 . *60'2] 

This proposition is used in the theory of the continuity of functions 
(*234-202). 

*604. H:/3eCl'a.7C/3.D. 7 eCl'a [*60-2 . *22"44] 
*60'41. V : e Cl'a .D.0nye Cl'a [*604 . *22'43] 

The following proposition is used in the theory of well-ordered series 
(*250'14). 

*6042. l-:/3 6Cl<a.7C/3. a !7.D.7eC]ex'a [*604'22] 



SECTION B] THE SUB-CLASSES OP A GIVEN CLASS 391 

*6043. H:& 7 eCl'a. = ./S^eCl'a [*22"59 . *60'2] 

*60 44. I" : e Cl'a . 7 e CI ex'a . D . £ u 7 e CI ex'a [#60-43 . *24'56 . *60'22] 

The following proposition is required in the theory of "first differences" 
(#170-65). 
#6045. H:peCl'(au£). = .(a7> 8).yeC\'a.&eC\'P .p = yv& 

Dem. 
V . #602 . #22-621-68 . D 

r-:peCl'(av/3).D.p = (pna)u(pn/3) (!) 

H . #602 . #2243 . D h . p n ae Cl'a . p n £ e Cl'/3 (2) 

h.(l).(2).*10-24.D 
h:peCl'(au/3).D.(a 7 , S).7eCl'a.SeCl'/3.p = 7uS ( 3 ) 

H.*60'2.D 

l-:(a7^)-7eCl f «.S6Cl'/3.p = 7 w S.D.(a7;S).7Ca.8C^.p = 7wS. 

[#2272] D.pCav£. 

[•60-2] D.peCl'(au/3) (4) 

I- . (3) . (4) . O H . Prop 

#605. Ks'Cl'a = a 

Dew. 

l-.#4011.#60-2.Dh:a?ea'Cl'o. = .(a^)./8Ca.a?e/8. (1) 

[#22-441] D.aea (2) 

K*22'42. Oh:a?ea. D.aCa. a?ea. 

[#10-24] D.(a£).£Ca.#e£. 

[(1)] D.a?6«'Cl'a (3) 

h.(2).(3).DH.Prop 

#60501. Ks'Clex'a = a 
Dew. 

K*40-ll.*60-21.Dh:#es'Clex'a. = .(a/3).£Ca«l!/3.*e#~- (1) 

[*22'441] O.aea (2) 

h . #22-42 . Df-:#ea.D.aCa.#ea. 

[*10'24.*24-5.*4-7] D .aCcc&la.xea. 

[*10'24] D . (a/8) . /3 C a . a ! /3 . a e £ . 

[(1)] D . a; e s'Cl ex'a (3) 

r- . (2) . (3) . D h . Prop 

The above proposition is used in the theory of cardinal multiplication 
(#11517). 
#6051. h.p'Cl'a = A [#40-22 . *60'3] 

The following proposition is used in the cardinal theory of finite and 
infinite (#124'541). 
#60 52. r-:s'*C/3. = ./tCCl'/3 [#40-151 .*60'2] 



392 PBOLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#60*53. h : £ Cp'x . = . £ e p'Cl"/c 
Dem. 

h. #40*15. #60*2. Dh ;. £ Cp'*. = : 7 e*.D y . £ eCl' 7 : 
[*40;41.#6014] = :£e;>'Cl"A::. D I- . Prop 

#6054. \-.C\'p'K=p'C\" l e [#60*53-2] 
#6055. H:Cl?a = Cl'£. = .a = £ 

K #30*37. #6014. D I- :a = £.D.Cl-a = Cl'# (l) 

h . #3037 . D r : Cl'o = Cl'/3 . D . a'Cl'a = s'Cl'fi . 

[*60-5] D.a = £ (2) 

K(l).(2).DKProp 

#60*56. H:Clex'a = Clex f i S. = .a = ^ [Proof as in #60*55] 

The following proposition is used frequently. 
#60*57. H . * C CIV* 
Dem. 

K #4013. #60*2. Dh:ae«.D.oeClV* (1) 

I- . (1) . #10*11 . #22*1 . D I- . Prop 

#60*6. \-:xea.D.i'a;eC\ex'a [#51*2*161 . #60*21] 

The following proposition is used in connection with cardinal multiplication 
and with greater and less (#1 15*17 and #117*66). 

#60*61. (- . t"a C CI ex'a [#37*61 . #51*12 .#60*6] 

*60*62. h:x,yea.'D.i'a;vi'yeC\ex l a [*60*6*44] 
#60*7. KCl'aeCls 2 

Dem. 
>.#60^.DH:/8eCl'a. = .i8Ca. 

[*22*l.*20*l-3] =.(&<!>,+). ci = z(<l>l2). = 2 (f I z).y}rla;D x <l>la;. 

[#105] D.( a f).£ = 2(^!*). 

[#20*4] D./3eCls (1) 

r . (1 ) . #60*2 . (#60*03) . Z> H . Prop 
#60-71. KCls 3 = Cl'Cls [(#60-03)] 
#60-72. h.Cls 8 = Cl'Cls 2 [(#60*04)] 



*61. THE SUB-RELATIONS OF A GIVEN RELATION 

Summary 0/46I. 

The propositions of this number (except that *6r371'372'373 imperfectly 
correspond to 46037 1) are the analogues of those with the same decimal part 
in #60. Proofs are omitted, as they are exactly analogous to those in 460. 
There are very few subsequent references to ^he propositions of this number. 



*6101. R1 = AP{\ = P(PGP)} Df 

46102. Rlex = AP{\ = #(PGP.a!P)} Df 

46103. Rel 2 = Rl'(Rel f Rel) Df 

46104. Rel 3 = Rl'(Rel 2 tRel 2 ) Df 

4611. r:XRlP. = .\ = P(J2GP) 

46111. r : \R1 ex P. = .\ = P(PGP. a l.R) 

46112. h.Rl'P = P(PGP) 

46113. \-.mex'P = R(RGP.KlR) 

46114. r . E ! Rl'P 

46115. KESRI ex'P 

4612. r:PeRl'P. = .PGP 
46121. h : i2eRl ex'P.^.RGP. a !P 
461-22. H : R eRlex'P . = . R eRl'P . a ! R 

46123. h : R e Rl ex'P . .= . R e Rl'P - t'A 

46124. h . Rl ex'P = Rl'P - t'A 

4613. r. A e Rl'P 
46131. H . a ! Rl'P 
461-32. I- . Rl'A = t'A 
461-321. h:P = A. = .Rl'P = t'P 
461-33^. KRlex'A = AnRel 

46134. KPeRl'P 

46135. HgjlP.D.P eRlex'P 

46136. r : a ! P . D . a ! Rl ex'P 
461361. I- : a ! P . = . a ! Rl ex'P 
461-362. KRl'(>4,y) = fc'A w'^y) 
461*37. r . Rl ex'O I y) = i'{x I y) 
461371. I- : R e 2 . D . Rl'P = t'A u t'JZ 



394 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*61-372. h:#62.D.RKRC0 r w2 

*61373. r : R e 2 r . D . UVR CO r u 2,. 

*61-38. H : R e 2 . = . Rl ex'E = t'E 

*6139. V . Rl'O ^ y vy s J, w) = t'A u i'(a; I y) \j i\z |w)u i l (x ^ y w z I w) 

*61'391. h : P, Q e 2 . D . R1'(P vQ) = l'A w t'P u i'Q sj t'(P c; Q) 

*61-4. \-:Q e m'P.RGQ.D.Ren\'P 

*61-41. h-QeRl'P.D.Q/SiZeRl'P 

*61'42. h-.Qe Rl'P .RGQ.ftlR.'D.Remex'P 

*61'43. h:Q, J ReRl'P. = .Qe/E e Rl'P 

*61-44. h:QeRl'P.12eRlex<P.D.Qc/i2eRlex<P 

*61-5. h . s'Rl'P = P 

«61-601. H . s'Rl ex'P = P 

*61-51. h.p'Rl'P = A 

*61-52. hs'XGQ.E.XC Rl'Q 

*6153. h : Q G p'\ . == . Q e jp'Rl"\. 

*61-54. l-.Rl*p'X=jp'Rl"\ 

*61'55. H.R1'P = R1'Q. = .P = Q 

*61-56. H . Rl ex'P = Rl ex'Q . = . P = Q 

*61'6. h : «Py . D . a? 4 y e Rl ex'P 

The analogue of #60*61 is not given, because we have no suitable notation 
for expressing it. 

*61'62. I- : xPy . zPw . D .ac \,y vy z ^w e Rl ex'P 

*617. h . Rl'P e Cl'Rel 



*62. THE RELATION OF MEMBERSHIP OF A CLASS 

Suminary o/#62. 

When "x e a" was denned, in #20, it was defined as a propositional 
function; and this mode of definition was necessary, because we had to treat 
of this function before treating of relations. But for many purposes it is 
desirable to regard e as a relation, so that "x e a" becomes an instance of the 
notation "uRv." This requires, strictly speaking, a change in the meaning of 
"x e a," but it is a change which does not falsify any of the previous propositions 
in which "x e a" occurs ; for if we call the new meaning "x e' a," i.e. if we put 

e'=xa{xea) Df, 
we have V : x e a . = . x e a. 

Hence it is unnecessary in practice to have a new notation for the new 

meaning, and we put simply 

e=<Kci(x e a.) Df. 

This definition, though strictly incorrect, is recommended by its convenience, 
and by the fact that it cannot lead to any harmful confusions. The new 
meaning of e may be taken as replacing the old throughout the remainder of 
this work. 

The uses of the propositions of the present number occur almost ex- 
clusively in the theory of selections from a class of classes (#83, #84, #85 and 
#88). Such selections are effected by means of selective relations, part of 
whose definition is that they are contained in e. Hence the uses of the present 
number. If tc is the class of classes from which a selection is to be made, a 
selective relation will in fact be contained in e f k; hence the properties of e [ k 
become important. Some of these properties are given in *62*4 ff. 

The most important propositions of the present number are the following: 
— * 
*62-2. h . e 'a = a 

*62'231...h :KCa'6. = .A~e« 

*62'26. \-.R = e\~R 

*62'3. h . e"/c = s'k 

*62*42. b: A^e k .D .d'e[K = k 

*62'43. \-.~D'e\-K = s'/c 

#62-55. r-:/cCl.D.ef*=iP* 



396 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*6201. e = Zca(ccea) Df 

.$621. \--.xea. = .x € a [*213 . (*62 01)] 

In the above proposition, the first e has the newly-defined meaning, while 

the second has the old meaning. In virtue of the above proposition, the new 

meaning may be substituted for the old in all propositions hitherto proved 

concerning e, and may take the place of the old meaning in all that follows. 

— > 
*62'2. Ke'a = a 

Dem. 

K*32-13.DKe'a = £(>ea) 
[*20-42] = a.Dh.Prop 

*6221. h.V^ = a(«ea) [*32131] 

Thus e'sc consists of tb.e classes of which # is a member. 

*6222. h.D'e = V 

Dem. 

K*24\L04.Dh.(».#eV. 
[*10-24] 0\-:(x):(^a).xea: 
[*33-13] Dh.(».#eD'e: 
[*2414] Dh.D f e = V 

*6223. K(Fe = Cls-t<A 

Dem. 

H . *53'5 . D h : aeCls- t'A . = . g ! a . 

[*33'131] = .aeCFe:DKProp 

*62231. h:/cCa'6. = .A~e« [*24'63 . *33'131] 



$62-24. 


k 


e|e=V 






Dem. 














h . *24104 


. *ll-57 


. D h . {x, y) . x e V . y e V . 






03111] 




D 1- . (x, y) . x e V . V e y . 






010-24] 




D h : (x, y) : (ga) . x e a . a e y : 






034-1] 




D h : (x, y) : x e | e y : 






025-14] 




DKe|e=t 


*6225. 


h 




(«"/?)} 




Dem. 











I- . *341 . *31 -11 . D I- : a (e | e) /3 . = . (ftx) .xea.xefi. 
022-33] = . a ! (a n £) : D I- . Prop 



SECTION B] THE RELATION OF MEMBERSHIP OF A CLASS 397 

#6226. h . R = e | R 
Dem. 

h . #3218 . D h : xRy . = .xeR'y. 
[*30-33.*321 2] = . (g«) . x e a . <xRy . 

[#341] = . x(e \R) y : D h . Prop 

#623. \-.€ (( k = s'k 

Dem. 

H . #371 . D h . e"« = £ {(go) .06«.«eo} 

[(*40'02)] = s<« . D H . Prop 

#6231. > . e 2 '* = s'/e 

Note that, since e is not a homogeneous relation, i.e. not one in which 
referent and relatum belgng to the same type, e 2 is strictly meaningless. 
For if we have x e a . a e k, the two e's have different meanings, and do not 
therefore properly give xe^K-. But it is convenient to allow e 2 , on the under- 
standing that the ambiguity of e is to be differently determined for the two 
factors in the product e| e, namely the second e must make both referent and 
relatum belong to the next type above that to which they respectively belong 
for the first e. 

Dem. 

h . #3213 . h . € zt K = x(xe 2 /c) 

[#345] =£{(aa).#ea.ae#} 

[(*40'02)] = s'k 

#6232. Ks = ee = e 2 [*30'41 .*62-3-31 . #37 1.1 J 

#6233. r-.T^JfCls 
Dem. 

h . #62-2 . #30-3 . D V : £ e a . = fi . = a . 
[#20-41] =p.,8 = a.aeCh. 

[#501. #35101] = p . (If Cls) a : D r- . Prop 

The use of #20-41 in the above proof depends upon the fact that a is 
merely an abbreviation for an expression of the form ^(^z). 

#62-34. KPe = sg'(P|e) 
Dem. 

h . #37-101 . (#37-01) .31-:. aP e /3 .=:«=£ {(ay) . y ej3 . xPy) 

[#341] =%{x(P\e)/3}: 

[#321-23] =:a{sg'(P|e)}/3:.DI-.Prop 



398 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#624. b.efic = xa(xea.aetc) [#21*2 . (#35-02)] 

The relation ef k is very important in cardinal arithmetic, in connection 
with the problem of selection from the members of k, i.e. of extracting one 
term out of each of the members of tc. A relation which is to effect this 
selection must be contained in e [ k. 

*6241. Y.a<e\K = K-i<A 

" Dem. 

\- . #35101 . D h : x(e[ k) a . = . x e o . a e k : 

[#1011-281] D H :. fax) . x (e \ tc) a . = : fax) . x e a . a e k : 

[#10-35] = : fax) .xeaiaetc: 

[#24*5] = : 3 ! a . a e /c : 

[#53*52] =:ae/c-i'A (1) 

K(l). #33-131. Dh. Prop 

*62'42. h:A~6/e.D.a'e !"«; = « 

Dew. 

h . #51-36 . D H : Hp . D .«: C - t'A . 

[#22-621] D.* = /e-fc'A. 

[#62-41] D . d'e T « = k : D H . Prop 

#6243. KD'e fK = sSe 

Dem. 

h .#3311 .Dr- .D'ef ,c = ti{faa) .x( € [ *)a} 

[#35-101] = x{faa).xea.ae/c} 

[(#40-02)] = s'k . D H . Prop 

#62-44. I- : R G e . = . (a) . R'a C a 

Dew. 

H . #231 . D H :. jB G e . = : #lta . D^ a . x e a : 

— > 

[#32-18] =:iB6iJ ( a.D, ]e .a;e«: 

[*ll-2.*22-l] =:(a).^oCo:.Dh. Prop 

#62-45. h :. R G e . E !! R"<I<R . = : a e CPtf . D a . #<a e a 
Dem. 
h. #14-21. #4-71. Dh:.i2<a€a. = :E!i2<a.l2<aea: 
[*30-33.*5-32] = :E!.R'a:a:.Ka.D a; .tfea (1) 

I- . (1) . #10-413 . D I- : : « e d'R . D a . R'a e a : = :. 

a e a'ii . D a : E ! i£< a : #ifa .D.^eo:. 
[*10-29.*ll-62] = :. a e d'R . D a . E I R'a : a e CPE . a£a . D«, * . * e a : . 
[*33-14.*4-7l] =:.ae(I'R.D a .ElR<a:xRa.D atX .xea:. 
[*37-104.*ll-2] = :. E !! R«a'R .Rde-.zDb. Prop 
This proposition is useful in the theory of selections. It is used in the 
proof of #83-27, and thence of #83-28. 



SECTION B] 



THE RELATION OF MEMBERSHIP OF A CLASS 



399 



#625. I- . i G e 

Bern. Y . #33 21 . #62-13 .OH. (Tt = 1 . 

[#52-1 73] 3 h : a e d' i\ 3« . t'a e a : 

[#62-45] D h . Tg e 

#6251. h:E!7'a.3.t'a = e'a 

Dm. H .#5215-172 . 3 H :. Hp . 3 : t't^a: 

[#51-15] D:«=t'a.= a ..fl5ea: 

[#303] 3 : T'a = e'a : . 3 K Prop 

#6252. h:E!e'a. = .ael. = .E!tV 

Dem. h. #30-2. 3 h:.E!e'a. = :(g&):#ea. =*.# = &: 

[#5211] =:ael: 

[#52-15] = :E!t'a:.3KProp 

#62-53. HESe'aO.e'a^'a [#62-51-52] 

#6254. h : a e 1 . 3 . e'a =^'a [*62*51*52] 

#62-55. h:/e.C1.3'.ef k ic = 7f k * 

Dem. V . #6254 . 3 V :. Hp . 3 : a e/c . 3 a . e'a = t'a : 

[#35-71] 3:ef/c=Tr*:3KProp 

#62-56. h.e| k t"a = i'| k t"a = a1t 
.Dew. 
I- . #52-3 . #62-55 . 3 b . e f i"a = If t"a 

I- . #35-101 . #37-6 . 3 I- :. x (t \ t"a) /8 . = 
[#51-51] = 

[#10-35] = 

[#13193] = 

[#51-23] = 

[#13195] = 

[#51-51] = 

[#351] = 

r.(l).(2).3h.Prop 

#62-57. h . T= € |" 1 

Dem. h. #62-55. 3 h. 6^1=^1 

[#5213] =1\QH 

[#35-452] =t.3h.Prop 



(1) 

xip'-(w) -v ,ea ■ £=*'» '•• 

= i t x:(>&y).yea.p = i<y: 

(3#) • = i t x.yea.i t x=t t y: 
(fty).& = i'x.yea.x = y: 
j3 = i l x . x e a : 

xifi .xea: 

■as (ail) (2) 



*63. RELATIVE TYPES OF CLASSES 

Summary of #63. 

The notations introduced in this and the two following numbers serve to 
express the type of one variable in terms of the type of another. They are 
very useful in arithmetic, where it is necessary to take account of types in 
order to avoid contradictions. The two chief notations are "t 'a," for the 
type in which a is contained, and "t'x" for the type of which a; is a member. 
We put 

#6302. <o ( a = au-« Df 

This defines "the type of members of a," or "the type which is of the 
same type as a." The characteristic of^ a type is that if t is a type, we have 

(a?) .xer, 
and conversely, if (x) . x e t, then r is a type. For in that case, "x e t" is true 
whenever it is significant, i.e. whenever x belongs to the type which is the 
range of significance of a; in "xer." Consequently t is this range of signifi- 
cance, i.e. is a type. 

Since we have (x) . x e (a u — a), it follows that a u — 'a is a type. It is 
not "the type of a" but "the type of the members of a." (In case a is null, 
"the type of the members of a" may be interpreted as meaning "the type to 
which x belongs when *xe a' is significant.") "The type of a?," i.e. the type of 
which a; is a member, is defined as follows: 

#6301. t'x = v'x w - i l x Df 

By what was said above, "t 'i'x" is the type of the members of t'x, i.e. the 
type of x. By combining the definitions of t'x and t 'a, we obtain 

h . t'x = to'i'x. 
Thus H . x € t'x and h : y =f= x . D . y e t'x. 

In short, t'x consists of everything either identical or not identical with x> 
that is, every y for which there is such a proposition, whether true or false, 
as "y = x." We put "t'x" here instead of "t'a," because x need not be a class, 
and is in fact subject to no limitation whatever, whereas "t 'x" is not signi- 
ficant unless a? is a class, and therefore we write "t 'ot" rather than %'«?." 

We put also 

*63011. t 1 'x = t'x Df 

This definition serves merely to bring t'x notationally into line with t 'x 
and the types P'x, t 3t x, . . . t 2 'x, t 3 'x, . . . defined below. 

In virtue of #20*8, we have 

h:^)«v~^a.D.^ (<£# v ~ <j>x) = t'a, 



SECTION B] RELATIVE TYPES OF CLASSES 401 

i.e. if "<f>a" is significant, then the range of significance of the function <f>z is 
the type of a. It follows that two ranges of significance which overlap are 
identical, and two different ranges of significance have no member in common. 

It will be seen that i l x is always of the next type above that of x, and s 1 k 
(if k is a class of classes) is of the next type below that of k. We put 

*6303. tfic^to's'K Df 

so that t^K is the type next below that in which k is contained. Thus if k is 

a class of classes of individuals, t^x is the class of individuals. We put also 

*6304. t*'x = t<t'x Df 
*63041. V'x = t'P'x Df and so on 
*6305. U'k^Wk Df 
*63 051. t 3 l K = tftfic Df and so on 

Thus given any two objects which are members of any one of the follow- 
ing: the type of x, the type of the classes to which x belongs, the type of the 
classes to which these classes belong, and so on, we can express the type of 
either of our two objects by means of its relation to the other object. 

The propositions of this and the two following numbers will hardly ever 
be used until we come to cardinal arithmetic. They are used constantly in 
the first section on cardinal arithmetic, and they are constantly relevant in 
the first section on relation-arithmetic. Moreover they are usually required for 
cardinal and ordinal existence-theorems. 

Among the most useful propositions of the present number are the 
following: 

*63 103. h . x e t'x 

*63105. h.aO 'a 

*6311. \-:xet 'a.D.t t x = a\J-a = t 'ct 

I.e. if x either is or is not a member of a, then the type of x is the type 
which contains a. This proposition uses *20-8. 

*6313. h : <f>x . (f>y . D . y e t l x 

I.e. if there is any function satisfied by both x and y, then y is of the type 
of x. It is necessary to the use of this proposition that, if <j>z is a typically 
ambiguous function, it should receive the same typical determination for x and 
for y. For example, we have always x = x and y = y; but we must not regard 
these as values of one function 2 = 2, because such a function is typically 
ambiguous. On the other hand, x = a and y = a are values of one function 
z= a, because here the presence of a renders the function typically determinate. 

*6315. \-.t 't'x = t'x 
*6319. I- . t'to'a = t'a 

B&w I 26 



402 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#6316. b:xet'y. = .yet'x. = .Rlt'xr\t'y. = .t'x = t'y 

This proposition, which depends upon #63*11, and thence upon #20*8 and 
*13'3, and thence upon #91415, is vital to the whole theory of types. 

#63*32. \-.t 1 f K = s%'/c 
#63-371. b:/3Ct <a. = .l3et'a 
#63383. b . t%'/c = to f K 

We shall have generally t»*V"ic — t m+n( K, where we may count suffixes as 
^negative indices, so that t m %'ic = t"^ ni K or t^ m f K according as m or n is the 
greater. 
#63-5. b : x € t 'ct . = . a e t*'x . = . a C t'x . = . t'oc = t 'a 

This proposition is used constantly. 
#63-51. I- : a e t '/e . = . aCtfie . = . tcCt'a .= . t'a = t '/c 
#63-52. b : aetfX . = . aC t a '\ . = . \Ct*'a. = . t'a^t^'X . = . V'a^U'X 

#63*53. H : * € t 'a ■ = ■ * 2 '# = #<*.= . t'x = V« 

The above fourpropositions, together with four similarones(*63-54-55-56-57), 
give transformations which enable us to express any relation of type, as be- 
tween class and members or members of members or etc., that is likely to 
occur in practice. 

#63-64. b.t'p = t 'i"$ 

This proposition is often used in the first section on cardinal arithmetic. 

#63-66. b.C\'t'x = t*x 



#6301. t'x = i'xv- t'x Df 

#63-011. t u x = t'x Df 

#63*02. Ca = aw-« Df 

#6303. Kk^Ws'k Df 

#63-04. t*'x = t't'x Df 

#63041. t 3 'x = t't*'x Df 

#63-05. U'ic = t x %'K Df 

#63051. U'k^W* Df 

#631. b.(x).xet 'a [#22-88] 

#63101. b . t'x = to'i'x = i'xv- i'x [#20-2 . (#6301-02)] 

#63102. b.{y).yet'x [#63-1101] 

#63103. b.xet'x [#63101 . #51-16] 

#63104. b:<f>x.~<f>y.0.yet'x [#63101 . #1314] 

#63105. h.aC^a [#2258] 

#63106. b.t 'a = t '-a [#228] 



SECTION B] RELATIVE TYPES OF CLASSES 403 

*63107. h :.(*). 4>x :f(<f>y) : D . <j>y 
Bern. 

h . *2 11 . *1011 . OK <y) ./(&/) v ~/(<^) (1) 

*- . (1) . *10'13-221 . D I- :.<*) . ^ . D : $y ./(&,) v ~/(<fa) : 
[*51] D : *y . = ./(^) v -/(&,) : 

[*2'2] D :/(</,y) . D . <f>y :. D h . Prop 

*63 108. I- :/(y e t l x) .D.yet'x [*63-107102] 

*63 109. I- :/(y e C«) .3.yet 'a [*631071] 

*6311. bzxet 6 'a.'2.t'x = a\J-a = t 'a 
Dem. 
h . *2234 .<*63'02) .D I- :. Hp . D : #ea . v . sc~ea : 

[*20'8] D :§(y€a. v .y ~ea) = #(y— #. v . y=|=^) : 

[*22-331.*5115] D:ou-«=( ( «u>6 (1) 

> . (1) . (*63'0102) .31-. Prop 

*63"12. H :. ^>a? v ~ fyx , 3 : <f>y v **> (f>y . = y . y e t'x 
Dem. 

1- . *631 1 . *20-8 . D h :. Hp . D : t ( x = 2 (0s) u - £ (£*) : 
[*20-31.*22-391'392] D : y e *<# . = , . 0y v ~ <f>y :. D H . Prop 

*6313. hifa.fy.O.yet'x [*6M2 . Imp . Add] 

*63 14. I- : («) . x e a . D . *„<« = a [*2414\L7-24 . (*63"02)] 

*6315. I- . %H l x = t'x [*6314102] 

*63151. K* %'* = V« [*63*141] 

*63152. V.xetfVx [*6310315] 

*6316. V M .xe?y m = .ye#a!. = .'£\tfixf\1?y. = .t , a: = t t y 
Dem. 

i-.*63-101.*51'23.3hzxet<y. = .yet'x (1) 

K*63\L3. Oh-.i^.zet'x.zet'y.D.yet'x (2) 
H . *63103 .Ohzyet'x.'D.yet'x.yet'y. 

[*10'24] "D.ftlt'xnt'y (3) 

K(2).(3). Dhzyet'x.E-.nlt'xnt'y (4) 

\-.*63'103.'2t-:t t x = t c y.D.yet f x (5) 

h.*6313. DH:yd'a;.^e^.D^€^ (6) 
h.*63-13. ^hzxefy.zet'y.D.zet'x: 

[(1)] DH :#€*'#. seJ'y.D.seJ'a; (7) 

K(6).(7). DI-:.ye^a;.D:^€i'a;. = .^e^2/ (8) 

K(o).(8). 2\-:.y € t'x. = .t i x = t'y (9) 
K(l).(4).(9).DKProp 

26—2 



404 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*6317. h-.yet'x.zet'y.D.zet'x [#63*16] 

#6318. h . a ! t 'a [#1025 . #631] 

#63181. h : a C U'P .s.j8C £„'« • = ■ 3 '■ *«>'« « <o'/9 . = . * <a = V£ 
Dem. 

I- . #63-105 . Dh:4'a= C/3 . D . a C C/3 (1) 

h.*24-6. Dh:.«CC/3.3:a = i„')8.v.a!^-a ( 2 ) 

1-. #63151. Dh:a = V/8.D.C« = W (3) 
h . #63-11 . D I- : a? e C/9 . * e - a . D . *'ar = t '/3 . t'x = t Q '-a. 

[#63106] D.t 'a = t '/3 (4) 

I- . (2) . (3) . (4) . D h : a C * </3 . D . C« = t '/3 (5) 

K (1) . (5) . D h : aCto',8 . = . U'* = U'P (6) 

K(6)^|. ■D\-i0Ct o t a. = .t,'a = to t /3 (7) 

K #6311. D\-:x€t 'ccnt '/3.D.t'x = t 'cc.t'x = t '/3. 

[#13171] D.* 'a = C/3 (8) 

h. #63-18. Dh:Va = C/3.D.a!« '««C^ (9) 

K (8) . (9) . D h : g ! C« « * '/8 . = . to' a = t '/3 (10) 

K(6).(7).(10).DKProp 

#63-182. H : a C C/3 • P C C7 ■ 3 • « C C7 [*63'181] 

#6319. I- . t%'a = t'a 

Dem. 

V . #63-105 . #22-42 . D I- . a C *„'« . C« C Ca . 

[#63-13] D\-.ocet't 'a. 

[#63-16] D I- . Prop 

#63-191. V.to'aet'a [#63-103-19] 

#63-2. H : x e t 'a . a e t 'ie . D . t 2t x = t'a = U'k 
Bern. 

h. #63-11. Dh:Hp.D.t'«=<o'a.«'o = C* (1) 

V . (1) . #6319 . (#63-04) . D H : Hp . D . V'x = £'a = *V* : D h . Prop 

#6321. H : a C t'x . = . C« = t'x 

Dem. 

V . #63-181-15 . D I- : a G t'x . = . t 'a = t 't'x 

[*63'1 5 J = $'<•;: D K Prop 

#63*22. h:«C^. = .aeC«. = .«'« = * '« 

h. #63-103. D\-:t'x = t 'a.D.xet 'a (1) 

h.(l).#63 , 11.3l-:a;€<o'o. = .*'aj = e 'o (2) 

h . (2) . #63-21 . D h . Prop 



SECTION B] RELATIVE TYPES OP CLASSES 405 

*6323. t-:aCt'x.KCt'a.D.t 2 'a; = t<a = t 'K [*63-2'22] 

Propositions of the same kind as the above can obviously be extended to 
t 3 'x, etc. 

*633. H : (a) . a e k . D . (x) . x e s'k 

Dem. 

h . *101 .DHrHp.D.Ve*. 

[*40'221] D.s'k=V. 

[*24'14] D .(x). x e s'k : D h . Prop 

*63 31. H . s'(k v-k) = s'k\j- s'k 

Dem. 
K#40-171. D\-:.xes l (Kyj-K). = :xes l K.y .xes* -k (1) 

h . (1) . *22-88 . *63-3 . D h : xcs'k . v .xes' -k (2) 

h.*22-88. 1\-:xes'K.v .xe- s'k (3) 

h.(2).(3).*10-221'13.D 

h :. xcs'k .v .xes' — k zxcs'k . v .xe — s'k :. 
[(1).*5\L] Dh:.a?e*'(*w-*). = :a?e*'#c. v.*e-s'*:. 3 h . Prop 

Note that the use of #10-221 in the above proof depends upon the fact 
that x € s'k occurs both in (2) and in (3), so that these are both of the form 

f(x € S'k). 

*63 32. h . U'k = s%'k [*63-31 . (*63-02'03)] 
*63-321. I- . t x 'K = UX'k = t %'K 

Dem. 

h . *2 0-2 . (*63-03) . D h . ^%'k = t 's% l K 

[*63-32] =U%'k (1) 

[*20-2.(*63'03)] = t %'s'K 

[*63-151] =t 's'K 

[*20-2.(*63-03)] =t 1 'K (2) 

I- . (1) . (2) . D h . Prop 

*63 33. h : t 'K = t 'X . D . U'k = t x '\ [*30*37 . *G3-:32] 

*63 34. V . U't'a = t 'a = s't'a 

Dem. 

h . *63-32 . D h . tft'a = s't Q 't'a 

[*63-15] = s't'a (1) 

[*63-101] = s'(i'a u - I'd) 

[*63-31] =s'i'ayj-s'i'a 

[*53-02] = a u - a 

[(*63-02)] = * 'a (2) 

K(l).(2).DKProp 



406 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#6335. h:t'a = t'j3.y.t (j 'oL = t i) 'l3 [#30*37 . #63-34] 
#6336. h:t'K = t'\.3.t 1 <K = t 1 '\ [*63'35-33] 
#63-361. \-:t 'a=t '/3. , 5.t<a = t</3 [*30-37 . #63-19] 
#6337. H:Ca=W. = .«'a = ^ [#63-35-361] 
#63 371. V:pQt<<OL. = .$et'a 
Dem. 

V . #63181 . D I- : /3 C t 'a . s . t 'a = t '/3 . 

[#63-37] =.t'a = t'l3. 

[#63-16] s./SeJ'aOh.Prop 

#6338. hiae^.ic^/a.D. t'a? = £ <a = */* 
.Dem. 

K. #63-11. Dh:Hp.D.^ = Ca.^a = V/f (1) 

I- . (1) . #63-34 . D b : Hp . 3 ■ . t 'a = ^ V* 

[#63-151-33] =t 1 'te (2) 

r.(l).(2).>KProp 

#63-381. \-:x€t 1 ' K .D.t'x=t 1 'K 
Dern. 

r- . #63-38-105 . 3h:aet '/e.xea.D~. t'x = ^ : 
[*10-ll-23.*40-ll] D h : xes%<K . D . *'# = */* (1) 

H . (1) . #63-32 . D h. Prop 

#63-382. Kg!*/* [#63-18 ; (#63-03)] 

#63-383. KiV* = C* 
Dem. 

h . #63-38-18 . #10-11-23-35 . D H : a e C* ■ 3 . t%' K = *%<« 
[#63-19] =t'a 

[*63-ll] = *„<* (1) 

h . (1) . #10-11-23 . #63-18 . D h . Prop 

#63-384. \-:t 1 ' K = t 1 ( \.3.t 'fc = t '\.t' K = t'\ [*63'383-37] 

#63-39. r- : tSic = t.'X . = . t f K = U'X . = .t'tc = t'X [*63-33-384-37] 

#63-391. I- : t'x = t'y . = . t 2 'x = t 2 'y 

Dem. 

I- . #63-39 . D h : t 2 'x = t*'y . = . U't'x = ft t'y . 
[#63-1 5] = . t'x == t'y : D I- . Prop 

*63-392. h : tftc = t 2 '\ . = . t* K = t.'X . = . t '/c = t 'X 
Dem. 

\- . #63-39 . D I- : t 2 '/c = t 2 'X . = . t % ( tc = t %'X . 

[#63-321] =.t 1 ' K = t 1 'X (1) 

h . (1) . #63-39 . D h . Prop 



SECTION B] RELATIVE TYPES OP CLASSES 407 

*63'4. \-:aet Q t K.ic6t '\.'2.to t a = t 1 'K = t 2 '\ 

Dem. 

b . *63-38-18 . D b : Hp . D . t 'a = t^x . t 'fc = tfX . 

[*30-37.(*63-05)] D . t 'a = t x 'ic . ^%'k = t 2 '\ . 

[*63-321] D . t 'a = t^tc . tfic = tf\ : D b . Prop 

*6341. b.t%'\ = t 1 <\ 

Dem. 

b . *63-4-18 . *10-11'23'35 . D I- : k e t '\ . D . t%'\ = t% f K 

[*63'383] =to'K 

[*63'38-18.*10-ll-23-35] = */X (1) 

h . (1) . *63 18 .Db. Prop 

*6342. \-.t*%'\ = t<<\ [*30-37 . *63*41-383] 

*6343. b .tfV'x^Vx [*63'3415] 

*6344. h.^'a = <o'« [*63-43'34] 

It is obvious that the analogues of the above propositions will hold for 
t 3 and ts, V and t i} etc. We shall not prove these analogues, but if occasion 
arises we shall assume them, referring to the corresponding propositions for 
& and t 2 . 

*63 5, I- : x e t ' a . = . a e t*'x . = . a C t l x . = . t'x = t 'a. 

Dem. 

b . *63-15 .Dh:aC^. = .aC %'t'x . 

[*63-371] =.aet* ( x (1) 

b . (1) . *63*22 . D b . Prop 

*63'51. b'.d€t 'K. = .a.Ct 1 'K. = .KCt l a. = '.t i a=zt h t K 

Dem. 

b . *4-2 . (*6303.) . D b : a C V* - s . a C £ V* . 

[*63'371-19] =.aet%<s'K. 

[*4'2.(*63-03)] =.aet%'K. 

[*63'383] =.aet Q ' K (1) 

b . (1) . *63-5-22 . D b . Prop 

*63'52. Hioe^.s.aC^.s.XC * 2 'a . = . *'a = */X . = .f'a = t '\ 
Dem. 

b . *63-51 — . (*63-03) . D 

bzaet^X. E.aC^'s'X. 

[*63-321] = . a C ^ %'**>< • 

[(*63-03'05)] =.aC* 2 '\ (1) 



408 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

r . #63321 . D 

h : a e t x '\ . = . a e t %'\ . 

[#63*22] =.t l a = t % f \ 

[#63-321] =^'\. (2) 

[*63-391-41-42] = . Vol = V* ■ (3) 

[*6315-181] =.\Ci '« w «. 

[#631 5] = .\0 2 <a (4) 

r . (1) . (2) . (3) . (4) . D r . Prop 

#6353. h : x e tfa . = . t 2 'x = t'a. = . t'x = £ '« 

Dem. 

\- . #3037 . D h : t*'x = tf'a . D . t x 't 2 'x = t x 't'a . 

[*63-43'34] ?.t'x = t 'a (.1) 

h . #6319 .DF:^ = C«.3- * 2 '# = *'a (2) 

h . (1) . (2) . *63-5 . D H . Prop 

#6354. h : a e «/* . = . tja = t x '/c . = .t'a = U'k . = . t il a = t'tc 

Dem. 

f- . #30-37 . D h : £'a = £ '* • 3 ■ tft'a = Wk . 
[#63-34-321] . D.to'a-^h'K (1) 

H . *30'37 . D I- : t l a = t x l K . D . W« = t%*K . 
[#63-19-383] D . t'a = t ' K (2) 

r . (1) . (2) . #63-51-53 . D h . Prop 

#63 55. h : « e £ '\ . = . */* = t 2 '\ . = . U ( k = t x '\ . = . t l ic = t '\ . = . V'k = t'X 
[Proof as in #63'54] 

#63 56. \-:xet 1 '/c. = .t ( x = t 1 'K. = .t 2 'x = t 'K 

Dem. 

b . #63-321 . D h : x e t x l K .= .xe t 't x '/c . 

[#63*53] =.f- t x=t% i K (1) 

[#63-383] = V« (2) 

h . (1) . #6353 . D I- : x e t x l K . = . t'x = «/*/* 

[#63-321] = tfK (3) 

h . (2) . (3) . D h . Prop 

#63-57. I- : a e t x '\ . = . * 'a = ^A • = . *'a = *A - = ■ t*'a = <o*X 
[Proof as in #63-56] 

#63-61. h . £ 2 <# = t'i'ac [#63-19-101] 

#63-62. h:xe t 'a . D . t'x e £'a . £*Va; = £'a 

Dem. 

K #63-53.3 hiHp.D.* 2 '^ =t ( a. 

[#63-61] 3.t'i'x = t t a. 

[#63-16] D.i'xet'a-.Dh. Prop 



SECTION B] RELATIVE TYPES OF CLASSES 409 

#63621. r- :xea .D .I'xet'a .t l i'x = t'a [*63'62 . #63105] 

#63 63. I- : x e to'a . D . 1'1'x e t*'a . t'i'i'x = t 2t a 

Bern. 

\- . *63-101 . D I- . tH'x = tji'i'x . 

[*63-62] D h : Hp . D . *'a = Vi'i'a; • 
[#63-19] D.t 2 'a = t'L'i'x (1) 

I- . (1) . #63103 . D h . Prop 
#63-64. I- . t'/3 = t < i"0 

Dem. 

h. #51 -16. #37-62. 

h : a? e /3 . 7> .xei'x ,i l xe t"# . 

[#63-105-38] D . a; e tji'x . U'l'x = V*"£ . 

[#13-13] O.xet^l"^ (1) 

h. (1). #63-51. Dh. Prop 

#63-65. I- . Cl%'a = fa [*63'371 . #60-2] 

#63-66. h . CI'*'* = t*'x [#63-5 . #602] 

#63-661. h . tf'Cl'a = « 2 <a [#60-34 . #63105-53] 

#63-67. I- . C\%'k = Wk [#63-51 . #60-2] 

#63-68. I- . C\%'k = t x l K [#63-52 . #60-2] 



*64. RELATIVE TYPES OF RELATIONS 

Summary o/*64. 

In the present number, we introduce notations defining the type of a 
relation relatively to the types of its domain and converse domain, when 
these types are given relatively to some fixed class a. If R is any relation, 
it is of the same type as t 'T>'R t Wd'B. If T)'R and d'R are both of the 
same type as a, R is of the same type as t 'a f t 'a, which is of the same type 
as a f a. The type of t 'a T ^ a we cal1 *«>'«» and the type of t m 'a f t nl a we call 
t mn 'a, and the type of t m 'a f t n 'a we call t mn 'a, and the type of t m 'a f t n 'a we 
call t m n 'a, and the type of t mi a f t n 'a we call m t n 'a. We thus have a means of 
expressing the type of any relation R in terms of the type Of a, provided the 
types of the domain and converse domain of R are given relatively to a. 

The most useful propositions of the present number are the following : 
*64 16. bzRGto'afto'p.s.Re t'(t 'a t U'P) 
*64'201. h:RGS.O.Ret'S.t'R = t'S 
#64-231. r : R e t'Q . D . D'R « t'D'Q . d'R e t'd'Q . C'R e t'C'Q 

Here "C'R e t'C'Q" will only be significant if R and Q are homogeneous 
relations, which is not required by the rest of the proposition. When R and 
Q are homogeneous relations we have 
*64-24. \-:R€t'Q. = .C'Re t'C'Q . = . t 'C'R = t 'C'Q 

This proposition is useful in connecting ordinal and cardinal existence- 
theorems. 
*64-312. b . t™ ( x = t u 't'x - tn'V'x 

*64-5. r-.Rl'(< 'ot«o'/8) = <W«T*o'/8) = *'(at/9) 

This proposition is frequently used. It states that the class of relations 
whose referents are of the type of members of a while their relata are of the 
type of members of /3 (i.e. the class of all relations contained in t 'a f t '/3) is 
the type of t Q 'a \ U'$ and is also the type of a f (5. 
*64-55. \-:C'PCt 'a. = .Pe t^'a 
*6457. hzC'PCt'x . = .Pe t n 'x 

The propositions of the present number are mostly obvious, though formal 
proofs are sometimes not very easily found. The use of the propositions of this 
number occurs chiefly in the first section on relation-arithmetic and in the 
proofs of existence-theorems in ordinal arithmetic and the theory of ratio. 



SECTION B] RELATIVE TYPES OF RELATIONS 411 

#6401. *oo'* = *W«H'«) Df 

#64011. P'x=*1?(t'x1#x) Df 

#64012. f l *'x = t t (t'x^t s 'x) Df 

#64013. t 2U x = t'(t*x | t'x) Df 

#64014. *»'* = *'(«■'« ft"*) Df 
etc. 

#6402. ^'a = t'(t 'a f ^'a) Df 

#64021. t w 'a = i'^'a t A>'a) Df 

#64022. t n 'a = t'(t 1 '0L\t 1 'aL) Df 
etc. 

#6403. V'a-^'a.tt'a) Df 

#64-031. ^»'a = ^'o t «'a) Df 
etc. 

#6404. V« = *'(*'« H'«) Df 

#64041. V« = «'(«*« T*i'«) ^ 
etc. 

#641. Kafaetfoo'a 
Dem. 

K#212. Dh:o = «,'a.D.«t« = «o'«t«»' a W 

H . #35-9 . D h : a f a = £ 'a T V« • 3 • a = Ca : 

[Transp] D h : a=K'a . D. a f a=K'a f *>'« (2) 

I- . (1) . (2) . D H :.a=Ca .v . a$t 'a:D: af a=C« t *<,'«• v. a f a+<„'« t *o'a (3) 
h . (3) . #5115 . #63101191 . I> h : a f a = * 'a H'a ■ v . a | a 4= C« t C« (4) 
1- . (4) . #5115 . #63101 . (#64-01) . D h . Prop 

#6411. h . too'ct = Z'(a T a) [*641 . *63'16] 

#6412. b.alflet'ito'alto'P) 
Dem. 
r . #35-85-86 . #63-18 . D r : a \ /8 = C« ? V£ . = . a = V« . £-= V/3 (1) 
K (1) . Transp . Dh« = * <a . /3 = t 'j3 . D . a f /3 = Ca f t o '0 : 

[*63101.*5115] D r : a = t 'a . D . a f e t'(to'a f t o '0) (2) 

K, (1) . Transp . Dh:oH'«-3.«ti8 + (^«tW- 

[*63-101.*51'15.Transp] D.«ti8e<U'«tW) ( 3 ) 

h.(2).(3).DK.Prop 

#6413. l-.*WatW) = ^(at/S) [#6412 .#63-16] 
#6414. I- . (a?, y) . x {t 'a f «/£) y [#631 . #35-103] 
#6415. r- . (E) . R G <b'o | $,'£ [#64-14 .#25-14-11] 



412 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#6416. h:22GVa?'o'£. = .-Re<W«TV£) 
Dem. 
h . *2'11 . D I- : R = C« t C/3 ■ v . E=K'« f V£ : 

[#23-42] D h : £ = V« ? C/3 ■ # G t 'a | C/3 . v . R 4= £ 'a t C£ (1) 
I- . (1) . #6415 . #10-22113 . D 

(-:i2G^at^/3:i2 = CatW.EGC«TW.v.iS: + C«tW (2) 
h . (2) . *5-l . D 
k:.EC^aTC/3. = :^ = ^atW.^GCaTW.v. J R4=CatW: 

[*23'42] = : R = t < a f W • v . R + A>'« T W : • ^ h ■ Pr0 P 

By putting V"'« (where i and s are some index and suffix which have been 
defined) for a and t/'a for /3, the above propositions give results applicable 
to any of the types defined at the beginning of this number, because of 
Wet = tj'a. 
#64-2. \-:<&lRnS.D.S € t'R.t'R = t'S [#63-13-16] 

#64-201. \-:RQS.D.Ret'S.t'R = t'S 

Bern. 

h . *25-6 . D h :. Hp . D : R = S . v . 3 ! £-^.R : 
[*13\L4] D:,R = ^.v.i2 + 5:.DI-.Prop 

*6421. I- : xRy . D . R e *<(<<* t *'3/) 

Dem. 

r . #63-103 . #35103 .Dh.« (t'a> t *'y) y i 1 ) 

K(l). DI-rHp.D-alJBA^t^y) ( 2 ) 

b . (2) . *64'2 . D K Prop 

#64 22. r . R e t'(t 'D'R | C O'jB) [#64*16 . #63-105 . *35'83] 

#6423. V.t'R = t's't'R 

Dem. 

h. #63-103. #41 -13. DI-.i2Gs^jB (1) 

h . (1) . #64-201 . D I- . Prop 
#64-231. h : £ e t'Q . D . V'R e t'D'Q . d'tf e t'd'Q . C'jR e t'G'Q 

Dem. 
\- . #63-12 . D h ::. Hp . D :: a?i2y . x , y :. a% . v .~{xQy) :: 
[#10-28] D : : (ay) .xRy.D x :. (ay) .xQy.v. (ay) ■ - (*%) = ■ 
• [#5-63] 3 X : • (33/) • a% : . v : . ~ (ay) . *% : (32/) • ~ *% : ■ 

[#3-26] D a :■ (33/) ■ «% ■ v • ~ (33/) ■ *% 0-) 

h.(l).#33-13.Dh:.Hp.D:a;eD' J R.D a; .^eD < Qw-D < Q: 

[(#63-02)] D:D'i2C«o'D'Q: 

[#63-371] D-.D'Re t'D'Q (2) 

Similarly h : Hp . ^ .d'Ret'd'Q .C'Ret'C'Q (3) 

h . (2) . (3) . D h . Prop 



SECTION B] RELATIVE TYPES OF RELATIONS 413 

#6424. \-:Ret'Q. = .C'Re t'C'Q . = . to'C'R = t 'C'Q 

This proposition is only significant when R and Q are homogeneous 

relations. 
Dem. 
V . #6422 . #63181 . D h . R e t'(t 'C'R f to'C'R) . 
[#13-12] D h : V^-B = ^C'Q .D.Re t'(t 'C'Q | VC'Q) (1) 

h . #64-22 . #63-181 .Dh.Qe t'&'C'Q T */C'Q) (2) 

h.(l).(2).*63-16. D\-:t 'C'R = to'C'Q.D.Ret'Q (3) 

I- . (3) . #64-231 . #63-16-37 . D 
\-:Ret'Q. = . t 'C'R = CC'Q . = . C'R e t'C'Q : D h . Prop 

#64-3. h : *</« = C/3 . = . a e t'fi . = . t'a = *</9 . = .t 'a = C/3 

h . #30-37 . (#64-01) . DH:C«==C/3.3.«oo'a = *oo'/3 (1) 

h.*64-l. Dh:Ctf=C/3.D.af aeCyS- 

[#64-16] D.at«G*o'/9?V/3- 

[#35-9-91] D.aCto'13. 

[#63-181] D.t 'a=t '/3 (2) 

1- . (1) . (2) . #63-16-37 . D h . Prop 

#64-31. r . «»'a? = C^ [#63-15 . (#64-01-011)] 

#64-311. 1- . t n 'a = too%'a [#63-321 . (*64'022-01)] 

#64-312. r . t™x = t n 't'x = C* 2 '« [#63-15 . (*63'04) . (#64-01 4-011 -01)] 

#64-313. > . 4/« = «n%'a = 4o%'« [#63-321 . (#63-05)] 

#64-32. h : <«' a = k'£ . s . * u 'a = *„'£ , = . t w 'a = t M 'P . = . * m a = *»'£ . 

= . «»*a = «»'/8 . = .a€t'/3. = .t'a = t'@ 
Dem. 

\- . #64-313-3 .Dh:^o = V/8 ■ = • «%'« = WP • 
[#63-41-39] = .t'a = t'/3 

Similarly the other equivalences are proved. 

#64-33. \-:ae t 'fi . = . t n 'a = t^'/i . = . t w 'a = tn'/i . = . t n 'a = t^'/i . 

= .t w 'a = t u 'ix. = .t'ct = t 'fi 
Dem. 

h . #64-311-313 . D h : t a 'a = t^'/x . = . t w %'a = C^V . 

[#64-3] =.t%'a = t%'fi. 

[*63-383-41-55] = . t'a = t Q 'fi (1) 

Similarly the other equivalences are proved. 

#64-34. I- : a e «//* . = . t w 'a = <22> • = • * 1U « = *u V ■ = ■ *"'«= *«« V ■ = ■ * 2 ' a = *o'/* 
[Proof as in *64'33] 



414 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*64'5. h . R1'(V« f V£) = t'(t 'a f tffi) = t'(ct f 0) [*64-13'16 . #61-2] 
#64-51. V.cclye t\t'x f «fy) [*64'21 . #55132] 

*6452. tixeU'a.yeU&.^.xiyet'ittta^ts'ft) [#6311 .#64-51] 

#6453. h : a? 6 Vff - 8 C * '£ . D . (fc'#) J, 8 e *'(*'a t *'£) 
Dm. 

I- . #6451 . D h . (e'fl?) 4 S € f («'*'« f *<8) (1) 

K #63-62. DHHp.D.£Va; = i<a (2) 

K*63\L81;37. Dr : Hp. D .i'S = *<£ (3) 

K(l).(2).(3).DKProp 
This proposition is used in connection with cardinal addition (#110 - 18). 
#64-54. I- . m<(t 'a t U'ol) = C« = t\a f a) = </Rl'(a t a) 

, [*64'5 . #61-34 . *63105-11 . (#6401)] 
*6455. \-iC l PCt«<0L. = .Pet m 'a 
Bern. 

h . #35-91 . D H : C'P C ^'a . = . P C C« t *<>'« • 
[#64-54] = . P € y« : D 1- . Prop 

#64-56. h . Rl'(tti> t *'«) = *"'* 
Dem. 

H . #64-5 . #63-15 . 3 r- . Rl'(*'a; f t l x) = *<(*'# | t'x) 
[(#64-011)] = *"<#. 3 r . Prop 

#64-57. hrCPC^.E.Pef'a! [#64-56 . *35'91 .#61-2] 
#646. I- . t'P =m t (t^D t P t tfd'P) 

Bern. 

Y . #35-83 . #63105 . D K PC fo'D'P f */<I<P . 

[#64-201] Dr.*'P = *'(*/D<Pf V^'P) 

[#64-5] =Rl<(<o'I>'PT4'<I'P).:DKProp 

#64-61. Y : D'P e t'a . d'P.e t'fr. D . t'P=t'(a f 0) 
Dem. 

Y . #6316-35 . D f- : Hp . :> . %T> 4 P= */« . Vd'P = U'P • 
[#646] D . t'P = ffo'a t V/8) 
[#64-5] = t'(a t /3) : D h . Prop 

#64-62. Y : D'P € t'TPQ . d'P e t'd'Q . = . P e *'Q . = . i'P = t'Q 

Dem. 

Y . #64-61 . D r : Hp . D . #<P = *<(D'Q f d<#) 
[*64-5-22.#63-16] =t'Q (1) 
h . (1) . #64-231 . 3 Y . Prop 

#64-63. I- : D'P et'u . d'Pet'ft . = . t<P = t'(a f j3) . = .Pet\a \ 0) 
Dem. 

Y . #64-5 . D 1- : *<P = t'{a f £) . D . *'P = *<(* <a t U'&) • 
[*64-231.*35-85-86] D . D'P e t%'a . d'P € t% l $ . 

[#63-19] D.D'Pet'a.a'Pet'fl (1) 

h . (1) . #64-61 . #6316 . D h . Prop 



*65. ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS 

Summary o/*65. 

In this number we are concerned with definitions and propositions in 
which an ambiguous symbol is determined as belonging to some assigned 
type. If "a" is an ambiguous symbol representing a class (such as A or V 
for example), "a x " is to denote what a becomes when its members are deter- 
mined as belonging to the type of x, while "a(x)" denotes what a becomes 
when its members are determined as belonging to the type of t'x. Thus 
e.g. "Y x " will be everything of the same type as x, i.e. t'x; V (x) will be t't'x. 
Similarly if "R" stands for a relation of ambiguous type, such as A or V, 
R x will denote what R becomes when its domain is confined within the type 
of x; R{x,y) will denote what R becomes when its domain and converse domain 
are confined respectively within the types of x and y; R{x,y) will have the 
domain and converse domain confined respectively to the types of t'x and t l y ; 
with analogous meanings for R (x) and jB (x v ). Throughout this number, 
R and a do not stand for proper variables, but for typically ambiguous symbols. 

The notations of the present number are used in the elementary parts of 
the theory of cardinals and ordinals, i.e. in Part III, Section A, and in Part IV, 
Section A. The only proposition, however, which is much used, is 

*65 13. r : a = fi x . = . a = t'x a /S . = . a C t'x . a = ft 

Here ft is supposed to be a typically ambiguous symbol. The first 
equivalence, "a = ft x .= . a = t'xnft" merely embodies the definition of ft x 
(#6 5 '01). It is the second equivalence that is important. Let us, for the 
sake of illustration, put 1 in place of ft. Then we are to have 

a — t'x r\ 1 . = . a C t'x . a = 1. 

(Since 1 is a class of classes, we shall have to suppose that x is a class.) 
Consider yea. If a = t'xr\ 1, yea. = .yet'x.yel. But we have (y) . y e t'x. 
Hence yeo.s.yel, whence a = 1. Also if a = t'x r\ 1, of course a C t'x. 
Thus a — t'xr\ l.D.aC<'a;.a = l. The converse implication follows from 
#22*621. The reason for the proposition is that a symbol such as "1," if it 
occurs in such a proposition as a = t'xr\l, must, for significance, be deter- 
mined as meaning that 1 which is of the same type as a, i.e. the class of all 
unit classes which are of the same type as members of a. And similarly, 
when we put a = 1, that does not mean that a is the class of all unit classes, 
but only that it is the class of all unit classes of the appropriate type, which 



416 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

if a C ('«, will be t'xnl. The proposition "^nl=l" is true whenevei it 
is significant, but t'x n 1 is typically definite when x is given, whereas 1 is 
typically ambiguous. The use of the above proposition lies in its enabling us 
to substitute typically definite symbols for such as are typically ambiguous. 

Another useful proposition is 

#652. h.sg'{R {x>y) } = R(xy) 

Here R is supposed to be a typically ambiguous symbol; the proposition 
states that if R is typically defined as going from objects of type x to objects 

of type y, then R must go from objects of type t'x to objects of type y. This 
proposition is only used twice (*102'3 and #154-2), but both uses are of great 
importance, the one in cardinal and the other in ordinal arithmetic. 

The only other proposition of this number which is subsequently used is 

#65 3. h . R p "fi = (R"fi)fi = R"h> n t'0 
This proposition is used in *102'84. 



#6501. a x = ar\ t'x Df 

#65 02. a O) = a n t't'x Df 

#6503. R x = (t'x) J \R Df 

*6504. R{x)^{t i 'x) J \R Df 

#651. R {x , „> = {t'x) 1 R \ (t'y) Df 

#6511. R{x y ) = {P'x) J [R\{t'y) Df 

#6512. R(x,y) = {V'x) J \RX(t i 'y) Df 

#6513. h : a = /3 X . = . a = t'xr* /3 . = . a C t'x . a = j3 

Bern. 

h . #4-2 . (#6501) . Dh:a = /3 iC . = .a = ^n/3 (1) 

K #22-621. #1313. Db:aCt'x.a = /3.D.a = t'xnf3 (2). 

H. #22-43. Dh : a = t'xn/3. D. «C^.«C . (3) 
[#63-13] D.&e t't'x. 

[#63-371-15] D.&Ct'x. 

[#22-621] O.fl^t'xnP (4) 

"h.(3).(4). Dh:a = ^ft j 8.D.aC^.a = /3 (5) 

h . (1) . (2) . (5) . D h . Prop 

#6514. \-:xet 'a.D.y(x) = y a [*63'53 . (*65-01-02)] 

#6515. h :xet 'a . D . R(x) = R a . iJ (*?„) = £<.,„, [#6353. (*65-03'04- 111)] 

#6516. b:xet 'a.y€t '/3.O.R(x,y)=R(x p )=R {a ^ [*63'53 .(#651-1112)] 



SECTION B] ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS 417 

*65-2. Ksg'fE,*,,,,} = £(*,,) 
Bern. 
K*32\L-23.(*65-l).D 

I- : a j>g'{jB (a . ( y) }] w . = . a = 2 {z e t'x . w e t'y . zRw} . 
[*22-39.*20-42] =.o = ^ftt(we t'y . zRw) . 
[*6513] = .aCt'x.a = ^{wet'y.zRw) (1) 

H . *20*33 . D h :: a = £ (w e t'y . zRw) . = :. z e a . = z . w e t'y . zRw :. 
[*63'108] = :.wet'y:zea.= z .wet'y,zRw:. 

[*4'73] = :. wet'y :zea . = z .zRw\. 

[*2033.*321] =:.W€t'y.aRw (2) 

K. (1) . (2) . *63-5 . D I- : a [sg'{i2 (a! , „,}] w.^.aeV'x.wet'y . aRw . 
[*35'102.(*65-11)] = . a \R{x y )) «;Oh. Prop 

*65'21. b.R (Zty) = {R {x> y) } te> y) 
Bern. 

K . *21-2 . (*65'1) . D r . {E^} te> „ = p« 1 [t'x 1 1* f *<</} \ t'y 
[*35-3334] ^t'x^R^t'y 

[(•651)] =2^.31-. Prop 

*65-22. H.-B(*y)«{-»(«.y)K«W) 

This and the following three propositions are proved as *65'21 is proved. 
*6523. *--R(x y )={R(x y )}(x y ) 
*6524. \-.R x = (R x ) x 
*6525. h.R(x)={R(x)}(x) 
*653. h . ^"/a = {R"p)ft = 22"/* « t'P 

Bern. 

h.*37-l.(*6503).Dh.W = ^{( ay ).y €At . a; %. a ; € ^} 

[*22-39.(*37'01)] =R"v,*t'& (1) 

■[(•65-01)] = (£"/*)/» (2) 

h.(l).(2).Dh.Prop 



B&w i 27 



SECTION C 
ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 

Summary of Section C. 

In the present section we have to consider three very important classes of 
relations, of which the use in arithmetic is constant. A one-many relation is 
a relation R such that, if y is any member of d'R, there is one, and only one, 
term x which has the relation R to y, i.e. R'y e 1. Thus the relation of father 
to son is one-many, because every son has one father and no more. The 
relation of husband to wife is one-many except in countries which practise 
polyandry. (It is one-many in monogamous as well as in polygamous countries, 
because, according to the definition, nothing is fixed as to the number of relata 
for a given referent, and there may be only one relatum for each given referent 
without the relation ceasing to be one-many according to the definition.) The 
relation in algebra of x 2 to x is one-many, but that of x to a? is not, because 
there are two different values of a? that give the same value of a?. 

When a relation R is one-many, R l y exists whenever y e G.'R, and vice 
versa; i.e. we have 

R e one-many . = : y e d'R . D„ . E ! R'y. 

Thus relations which give descriptive functions that are existent whenever 
their arguments belong to the converse domains of the relations in question 

are one-many relations. Hence Cnv, D, d, C, R, R, sg, gs, R e ,p, s, p, s, I, i, i, 
CI, Rl are all of them one-many relations. 

When R is a one-many relation, R'y is a one-valued function-; conversely, 
every one-valued function is derivable from a one-many relation. A many- 
valued function of y is a member of R'y, where R'y is not a unit class, and 
any one of its members is regarded as a value of the function for the argu- 
ment y\ but a one- valued function of y is the single term R'y which is 
obtained when R is one-many. Thus for example the sine would, in our 
notation, appear as a relation, i.e. we should put 

sm = ^{x^y-f/S\ + y s l5\-...} Df, 
whence sm'y = y-y 3 /S \+y 6 /5 !- ..., 

so that "sin'y" has the usual meaning of sin?/. Then instead of sin -1 #, we 

should have sin'#, which would be the class of values of sin -1 x; and instead 
of "y = sin -1 a;," which is a misleading notation because y = sm~ 1 x and 

2 = sin -1 a: do not imply y = z, we should have ye sin 'a;. Similar remarks 
would apply to any of the other functions that occur in analysis. 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 419 

A relation R is called many- one when, if x is any member of D'i2, there 

is one, and only one, term y to which x has the relation R, i.e. R'xel. Thus 
many-one relations are the converses of one-many relations. When a relation 

R is many-one, R'x exists whenever x e D'R. 

A relation is called one-one when it is both one-many and many-one, or, 
what comes to the same, when both it and its converse are one-many. Of the 

one-many relations above enumerated, Cnv, sg, gs, I, i, i, CI, Rl are one-one. 

Two classes a, ft are said to be similar when there is a one-one relation R 
such that T> l R = a . Q'R = (3, i.e. when their terms can be connected one to 
one, so that no term of either is omitted or repeated. We write "asm/3" for 
" a is similar to yS." When two classes are similar, the cardinal numbers of 
their terms are the same; it is this fact chiefly that makes one-one relations 
of fundamental importance in cardinal arithmetic. 

According to the above, a relation is one-many when 

yed'R.Dy.R'yel, 

i.e. when R"(I<R C 1. 

Similarly a relation is many-one when 

^"D'E G 1, 

and a relation is one-one when both conditions are fulfilled. The classes 

R ee d'R, R'tD'R, which appear here, are often important; some of their 
properties have already been given in *37-77*77r772-773 and in #5361 to 
*53-641. 

It is convenient to regard one-many, many-one and one-one relations as 
particular cases of relations which, for some given a and /3, have 

R"<1'R C a .%'D'R C fi. 

We put a-+l3 = R{R"a'RCa. 4 R"D<RCl3} Df. 

Hence, without a new definition, " 1 — ♦ 1 " becomes the class of one-one 
relations; also, as will be shown, "1— »Cls" becomes the class of one-many 
relations, and "Cls— >1" becomes the class of many-one relations. Although 
it is chiefly these three special values of a— >/3 that are important, we shall 
begin by a general study of classes of relations of the form a — > /3. 



27—2 



*70. RELATIONS WHOSE CLASSES OF REFERENTS AND OF 
RELATA BELONG TO GIVEN CLASSES 

Summary of #70. 

If a and /8 are two given classes of classes, a relation R is said to belong 

to the class a — > $ if R'y e a whenever y e (I'R, and R'x e fi whenever x e T>'R. 
If only one of these conditions is to be imposed, this result is secured by re- 
placing the class involved in the other condition by "Cls," since "R'y eCls" 

< — 
always holds, and so does "i2'#eCls," and therefore neither imposes any 

limitation on R. In the most important cases, a and /S are either both cardinal 
numbers, or one is a cardinal number while the other is Cls. 

In virtue of *37*702'703, the conditions above mentioned as imposed upon 
R by membership of a — > (3 are equivalent to 

~R"<1'RQol.'R"T><RCP. 

This form is used in the definition (#70-01). 

The propositions of the present number are hardly ever used except in #71, 
where a and /3 are both replaced by 1 or Cls. The most useful propositions are 

#701. \-:R6CL-*p. = .R"a<RCa.R"WRQ(3 

(This merely embodies the definition.) 

— ► <— 

#7013. h:.jRea-*£.= :(y). R'y eau i'A:(x).R l x € 0u i'A 

#7022. K£->a = Cnv"(a->£) 

#70-4. \-.a-+Cte = R(R"a<RCa) 

#70-41. KCls->£ = £CR"D' J RC/9) 

#70-42. h . a -> /3 = (a -> Cls) n (Cls -► £) 

#70-54. h : d'R n a<S = A . R, S e a -* Cls . D . R vy 8 e a -> Cls 

with similar propositions for Cls — > £S and a — * /3. 
#7062. \-:Rea-*C\s.3.Rtvea^>C\s 

with a similar proposition for Cls — > fi. 

#7001. a-+{3 = R(R"a<RCa. 4 R"I)<RC/3) Df 

#701. r- : R € a -► /3 . s .l!"(I<i2 C a . ^"D'E C £ [#203 . (#7001)] 

— > «— 

#7011. h:.i2ea->£. = :ye d'R .2 y .R'yea:x€D'R.Dx-R'xeP 

[#37-702-703 . #701] 



CLASSES OF REFERENTS AND RELATA 



421 



y € V . Dj, . R'y e a w l'A : 
{y)~R'yeayJi'A (1) 

{x).*R'xe$vJi'A (2) 



SECTION C] 

*7012. \-:Rea->/3. = .R"VCavi'A.R"VC0yJi'A [*701 . *5362-621] 

— ► «— 

*7013. \-:.Rea-+0. = :(y).R'yeavi'A:(x).R'xePsJL'A 

Dem. 

b . *37*702 . D h :. E"V C a v fc'A . = 

[*24-104.*5-5] s 

Similarly 1- :. R"Y C^ut f A.= 

h . (1) . (2) . *7012 . D h . Prop 

— > — > *— tr 

*7014. b :: Rea->0 . = :.(y):R'yea.v.R'y = A:.(x):R'xe .v.R'x= A 

[*7013 . *51'236] 

*70 15. h :. R e a -> /9 . = : a ! R'y . D y . R'y e a : g ! R'x . D* . R'x e 
[*2451 . *4-6 . *7014] 

*7016. h : R e a ->£ . = . D<ft C a u t'A.D-^RC/Sut'A [*3778'781 . *7012] 

*7017. b :: Ae a. D :. Re a-* /3. = :(y). R'y eazftl R'x. 3 x .R'xe 8 
Dem. 

h . *51-2 . *22'62 .Dh:Hp.D.a = aui'A (1) 

h . (1) . *7013 . D 

H-.zHp.Dz.Eea-^iS.siC^.^i/eazC^.^e/gut'A (2) 

H . *51*236 . D h :. S~'#e/3 u t'A . = : R'xefi . v . R'x = A : 

[*24-51.*4-6] = :^\R'x.O.R'xe0 (3) 

h . (2) . (3) . D h . Prop 

*70171. h :: A e/3 . D :. E e a ->£ . = : 3 ! 5*y . D„ . R'y e a : (x) . R'xe 
[Proof as in *7017] 

*7018. h: ! Ae«.Aei9.D:.Be«->i9.5:(y).5'y€a:(*).S i *€i8 
[Proof as in #70*17] 

*70'2. 1- . a ->£ = (« w i'A)-»/3 = a -♦ (/8 w //A) = (a w t'A) ->(£ v t'A) 

Dem. 
h . *22-58-62 . DK(ou t'A) ui'A = aw l'A . (/3 w t'A) wt'A = j 8w i'A (1) 

h . *70i2 .(1) . Dh:.Bea->£. = . E" V C (a w t'A) u l'A . R"V C /3 u l'A . 
[*70-12] =.iie(owi'A)-»3. (2) 

[*70\L2.(1)] = .~R"VC(clvjl'A)vl'A.'r"VC(I3vi'A)vi'A. 

[*7012] =.E€(owi'A)->()9wt'A). (3) 

[*70\L2.(1)] = . ~R"V Cawi'A. *R"V C (£ u i'A) w t'A . 

[*70'12] =.R€a-*(/3yJi'A) (4) 

K(2).(3).(4).DKProp 



422 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*70-21. b.a-+@ = (a-i'A)-+/3=a-^(/3-i'A) = (a-i<A)-*(l3-i<A) 

Dem. 
h . #51222 . D h : A~e a . D . a - t'A = a : A~e . D . £ - i'A = /3 (1) 

I-. #51-221. Dh: A ea.D. (a- t'A)w'A = a: A e/3. D .(/3- t'A)u/3 = /3 (2) 
K(1).D 

h:A~ea.3.(cL-i<A)^>/3 = a-+/3.(a-i'A)^>(/3-i<A) = ci-+(j3-i<A) (3) 
I- . (2) . #70-2 . D 

h:Aea.D.(a-t'A)^/3 = a^/Ma-t'A)^(/3-t'A) = a-*03-t'A) (4) 
I- . (3) . (4) . #4-83 . D 

h.(a-t'A)^/3 = a^/M«-''A)-*0S-t'A)=a->(/3-i'A) (5) 

Similarly h . a -» (£ - t'A) = a -»/3 . (a - t'A) -+(/3 - t'A) = (a - t'A) -*£ (6) 
I- . (5) . (6) . D r . Prop 

#7022. h . /3 -> a = Cnv"(a -> £) 

Z)era. 
h. #376. #3113. D 



I- :.QeCnv"(a -ȣ).= 

[#70-12] 
[#32-24-241] 



[#13-193] 



: (gi2) .Rea-+fi.Q = Cnv'R : 
= : (gJB) . i?"V Caw t'A .5"V C £ u t'A . Q = Cnv'i? : 
= : (a- 8 ) ■ (gs'Cnv'i2)"V Caut'A. 

(sg'Cnv'22)"V C £ u t'A . Q = Cnv'iZ : 
s:( aJ R).(gs'Q)"VCav,t'A. 

(sg'Q)" V C /3 u t'A . Q = Cnv'E : 

[#32-23-231 .#10-35] = r V"V C a v t'A . Q" V C /3 u t'A : (gi2) . Q = Cnv'i? : 
[*31-33.*10-24] = : V"V C a u t'A . #"V C u t'A : 

[#70-12] =:Qe/3-*a:. D h . Prop 

#70-3. h.aCY.^CS.D.a-^^Cv-^S 

Dem. 
h . #701 .Dh:Hp.i2ea->/3.D .~R"<I'R C a . S"D'i2 Cj3.oiCy.j3C8. 

[#2244] D . i2"CI'i2 C 7 . J2"D'12 C S . 

[#701] D.^e 7 -»S (1) 

K (1) . Exp . #10-11-21 . D h . Prop 

#70-31. b.(a-*P).r\(y-*S) = (ar\y)-*(J3f\8) 

Dem. 
h.*70-l.DI-:i2e(a-*/8)n(7-»8). = . 

^"d'iZ C a . if' 'd'iZ C 7 . S"D'7£ C /3 . £"D'iZ C £ . 

[#22-45] = . ^"a'.R C a n 7 . i2"D'i£ C /3 n 8 . 

[#701] =. J Ke(«n7)-»(/3nS):Dl-.Prop 



SECTION C] CLASSES OF REFERENTS AND RELATA 423 

#7032. K(a->/3)u(7->8)C(au7)->(£u8) 

Dera. 
I-. #70-1. DI-:.-Re(«-* J 8 ) w (7 ->£)■ = ■ _^ <_ 

R«a<R C a .^"D'iZ C . v . E"(I<.R C 7 . IF'D'i* C 8 : 
[*3-26 2748] D :~R"(I<R C a . v . E"d<.R C 7 : ie"D<E C /3 . v . R«D'RC 8 : 
[#22*65 j D : JR"<I'.R Cau 7 . E"D<£ C /3 u 8 : 
[#701] D : R e (a u 7 ) -* (£ u 8) :. D K Prop 

#704. H . a -* Cls = R (£"<!' R C a) 

K*701OH:i2ea^Cls. = .^"a<iSCa.iJ"D'i2CCls. 

[♦87-761] =.WiJCo:DKProp 

#7041. h.Cls^£ = £(S"''D-\RC£) [Proof as in #70'4] 

#7042. K«->/3 = («-»Cls)«(Cls->£) [#70-4-41] 

*70'43. \-:.Rea-+Cte. = :yea<R.D y .R'yea [As in #7011] 

#70 431. h :. R e Cls -> £ . = : x e D'R . D* . R'x e £ [As in *70-ll] 

#7044. I- : ie e a -> Cls . = . jR"V Cawt'A [As in #7012] 

#70441. h : .R e Cls -> £ . = . £"V C w t'A [As in #7012] 

#7045. H-Eea^Cls.EE.^.Jfyeaut'A [As in #70-13] 

#70 451. H : E e Cls -> £ . s . (*) .%* e £ u t'A {As in #7013] 

#7046. \-i.Rea-+Cte.= :(y):R'yea.v.R'y = A [As in #7 01 4] 
#70-461. H :. R e Cls -*£. = : (a) : £'# e £ . v . £<# = A [As in #7014] 
#7047. \-:.R € a->C\s. = :Kl~R'y.3 y .'R'y€« [As in #70-15] 

#70471. \-:.ReC]B->0. = '.nlR'*.'}x'*R t xeP [As in #7015] 

#70 48. H : R e a -■» Cls . = . D'EC a w t'A [As in #7016] 

*7(H81. H : R e Cls -» £ . s . D'E C £ u t'A [As in #70*16] 

#70-5. H ■ Cls -> a = Cnv"(a -* Cls) . a -► Cls = Cnv"(Cls -> a) [#70*22] 
#7051. I- :. f , 17 e a . 3f,, . £ n 17 e a *-» t'A : D : .R, £ e a-*Cls . D . i2 n £ e a->Cls 

Dem. 
h. #32*3.3 \-:.Hv.3:R'yea.S'yea.'3.{sg'(R*S)}'yeavi'A (1) 

K #32*3. #51*1 5. #24*34.3 

h:^ea.^6t f A.D.{sg < ( J Rn^)} t 2/ = A. 
[#51-236] D.{sg'(i2nS)} 4 2/e««t'A (2) 



424 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



H.(l).(2).*4-4.DI-:.Hp.D:i2Va-'S'y€« wt 'A.D.{sg'(En>S')}Vavt f A (3) 
H . *32-3 . *5l 15 . *24'34 . *51236 . D 

I- : R'y e t'A . /Sty e a u t'A . 3 . {sg<(22 n ,Sf)}'y e a u i<A (4) 

h.(3).(4).*4-4.3h:.Hp.D:i2'y,^eout'A.D.{sg'(i2n < Sf)}'y€aut'A: 
[*10ir2P27.*7(H5] D:i2,£€a->Cls.D.(y). {sg*CR AS)}'yea ui'A . 

[*70-45.*32-23] D . £ r» £ e a -> Cls :. D h . Prop 

*70*52. H :. £ V€ p . D f „ . |« v e£ v *<A : I>:i2,£eCls->/3.D. J? A SeCla->/3 
[Proof as in *70'51] 

*7053. H. $, i;eaOf,,.£n 17 eavi'A: £176/8. D^.fniye/Svi'AO: 

Dem. 

V .*70-5-51 . D h :. Hp . D : R, Sea-+Ch . R,SeCte->/3 . Z> . 

EnSea-»Cls..Rn£eCls^i8 (1) 
K(l).*70-42.DKProp 

*7054 h:a^naSS = A.J?,#ea^Cls.D.i?<y#ea-*Cls 

h . *24'15 . *22-33 . D 

\-:.a.'Rn(I'S=AiD:(y):~{ye<I'R.yea<& r . 
[*3341] D 

[*4'51.*24-51] D 

[*24-36] D 

h.*70-45.D 

h : . R, S e a -► Cls . D : (y) . R'y eaui'A: (y) .^eawt'A 
I- . (1) . (2) . D h : . Hp . D : (y) .~R'y w 5*y e a u t'A : 
[*32-32] D:(y).{sg'CKe/S)}<yeaw'A: 

[*70-45] D:Eiy£ea-»Cls:.DI-.Prop 

*7055. \-:T>'RrxD<S = A.R,SeC\s->l3.D.RvSeCh->/3 

[Proof as in *70'54] 

*70*56. h:D<R*T) t S=A.a'R*a i S=A.R ) S€a-+0.3.RK>S € a->i3 
[*70-54-55'42] 

*7057. \-:C<RnC'S = A.R,Sea-+/3.1.RvSea-+/3 

Bern. 
F . *33161 . D h . D<£ n D'S C C"i2 n C'S . d'R n a<S C C'R r» C'S . 
[*2413] D I- : C'RnC'S=A . D . D'i£nD<£ = A . (Pi? n a<£ = A (1) 

K(l).*70-56.Dh.Prop 



(y):~{ a !i2'y. a !>Sf'y}: 

(y):~&y = A.v.~S'y = A: 

(y) : i2'y u S'y = S<y . v . R'y v&y = R<y (1) 



(2) 



SECTION C] CLASSES OF REFERENTS AND RELATA 425 

*70'6. \-:Sea-*C\s.R"<aCayji'A:'}.R\Sea-+Cte 
Bern, 
h . *37-31 . D H . {sg'CR i S)]"V = (Re \S)"V 

[•87-33] = R e <<8"V (1) 

h.(l).*70-44.DI-:iSeo-*Cls.D.{sg'(12|S)}"VCJBe"(awt'A) (2) 
h . *3722 . Dh. R € "(a u t'A) = R e "a u jR e ' VA 
[*5331] = R e "a « t'Ik'A 

[(*37-04).*3Ml-29] =R«'ayJi'A (3) 

h . (3) . *22-66 . D h : R"'a Caui'A.D. i2e"(a w t'A) Caui'Ay t'A . 
[*22-56] D . Re"{* u t' A) C a w t'A (4) 

h . (2) . (4) . D h : Hp . D . {sg'(i2 1 £)}"V Coyi'A. 
[*70'44] D.i^l^ea-^ClsOI-.Prop 

*7061. h:J2eCls-»/8.jS"'/8CiSwi'A.D.^|flf€Gl8-»/8 [As in *70'6] 
*7062. f-:i2€a-^Cls.D.i2p 7 ea^Cls 
Dem. 

V . *35-64 . Transp .3h:y~ey.D.y~e a'(R [ y) . 

[*33-41.*24'51] D . {sg'(JS [ y)}'y = A . 

[*51'236] D.{8g'(Rty)}'yeavi'A (1) 

I- . *35101 . *4-73 .D\-:.yey.D:x(R[y)y.= x . xRy : 

[*2015.*3213'23] D : {sg'(i2 f y)}'y = R'y (2) 

K*7045. DhiHp.D.jK'yeaut'A (3) 

K(2).(3). D\-:.Kp.O:yey.D.{sg'(R\-y)}<yeavi<A (4) 

H . (1) . (4) . *4'83 . D h : Hp . D . {sg'(R [ y))'y eaut'A (5) 
K (5) . *101121 . *70'45 . D h . Prop 

*7063. \-:ReCte->0.-D.81ReCte-*j3 [As iiMfc70-62] 



*71. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 

Summary o/"*7l. 

In this number we shall be concerned with the more elementary properties 
of one-many, many-one, and one-one relations. These properties are very 
numerous and very important. The properties of many-one relations (i.e. of 
relations belonging to the class Cls-*1) result from those of one-many rela- 
tions by means of *70'5, whence it follows that many-one relations are the 
converses of one-many relations. It is thus only necessary to interchange 

R and R, D and CE, R and R in order to obtain a property of a many-one 
relation from a property of a one-many relation. Or we may repeat the 
various steps of any proof, making the above interchanges at every step, and 
the analogous proposition will result. For this reason, in what follows, we 
shall omit all proofs of properties of many-one relations, confining ourselves to 
proving the analogous properties of one-many relations. 

In virtue of #70 - 42, one-one relations (i.e. relations belonging to the class 
1 — > 1) are the relations which are both one-many and many-one; hence their 
properties result from combining the properties of one-many and many-one 
relations. We shall omit the proofs when they consist merely in such 
combinations. 

A one-many relation gives rise to a descriptive function which is existent 
whenever its argument belongs to the converse domain of the relation. That 
is, if R e 1 — > Cls, we have E ! R l y whenever y e Q.'R. Conversely, if a descrip- 
tive function R'y exists for the argument y, then R- is one-many so far' as that 

— > 
argument is concerned, i.e. R'y e 1. Thus we find 

R e 1 -> Cls . = . E !! R^a'R. 

The descriptive function R ( y derived from a one-many relation R has thus 
a definite value whenever yeQ'R, and not otherwise. Thus the class of 
arguments for which such a function exists is the converse domain of the 
relation which gives rise to the function, i.e. 

Ee 1 -* Cls. D.£{E! 22<y} = <!<£, 
and the converse implication also holds. 

.It often happens that a relation which is not in general one-many becomes 
so when its domain, converse domain, or field is subjected to some limitation. 
For example, let R be the relation of parent to child, a the class of males, and 
/3 the class of females. Then R is not one-many, but a^| R and /?1 R are one- 
many, and in fact (a 1 R)'y = the father of y, (fi 1 R)'y = the mother of y. We 
shall often have occasion to deal with relations obtained by limitations imposed 
onDorQ; thus a (D f X) R . = . R belongs to the class X, and has a for its 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 427 

domain. The class \ may be so constituted that only one relation R fulfils 
this condition; in that case, DTXeCls^l. Since Del->Cls, we find 
DrA.eCls-»l.= .D| k \el-+l. This type of condition, D|"\el->1 or 
<ir\el->l or CfXel-frl, is one which frequently occurs in subsequent 
work. Another condition which often occurs is JPf\eCls-»l. When this 
condition is realized, a term x which belongs to the field of one relation of the 
class \ does not belong to the field of any other relation of this class, i.e. the 
fields of relations of this class are mutually exclusive. 

For purposes of realizing imaginatively the properties of one-many 
relations, it is often convenient to picture their structure as in the accom- 
panying figure. Here x, y,z, ... form the domain of R, and all the points 




R y 



z • 




in the oval marked R'x are such that x has the relation R to each of them, 
with similar conditions for y and z. What characterizes R as a 1 -* Cls 
is the absence of overlapping in the ovals. For if R'x and R'y had a point 
in common, this would be a relatum both to x and y, and both x and y 
would be referents to it; whereas in a 1 ->Cls, no term has more than one 
referent. 

The above figure illustrates a very important property of one-many rela- 
tions, namely 

Rel->Ch. = .R\R = I[T> ( R. 



428 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

In the above figure, / f D'R is the relation of identity confined to x, y,z,.... 
If R were not a 1— >Cls, we could sometimes go from a? to some term of 
R'x r\ R'y by the relation R, and thence back to y by the relation R. But 

when .Kel-^Cls, R\R must bring us back to the point from which we 
started. 

4— 4— 4— 

When J? el ->1, each of the ovals R'x, R'y, R% ... in the above figure 

4— v 

shrinks to a single point, so that R'x = t'R'x. Thus when R is given as a 
l-»Cls, it will be a 1->1 if R'y = R'z .3 y>z .y = z. This proposition is 
constantly used, and so is the consequence that R[ fi is a 1 — > 1 if 
y,zefi. R'y = R'z.D y>z .y = z. (These propositions are *7l-54"55 below.) 

The hypothesis R e 1 -» Gls is equivalent to the hypothesis 
xRz . yRz . D Xi y tZ • ® = y 
(cf. *71'17, below), and the hypothesis ReCls— > 1 is equivalent to 

xRy . xRz . ^ x ,v,z >y = z. 
These are for many purposes the most convenient hypotheses to use. 

The most useful propositions in the present number are the following. 
(We omit here propositions concerning Cls — > 1 or 1 — > 1 which are mere 
analogues of propositions concerning 1 — > Cls.) 

*7116. r : R e 1 -> Cls . = . E !! R"<1'R 

This gives the connection of one-many relations with descriptive functions. 
We have also 

*71163. f :. R e 1 ->Cls . = : y € (I'R .= y . El R'y 

For many of the constant relations defined from time to time, such as Cnv 
or D, the following proposition is useful: 

*71166. r- : (y) . E ! R'y . D . R e 1 -> Cls 

*7117. \- :. R e 1 -> Cls . = : xRz . yRz . D Xi y >z .x = y 

This might have been taken as the definition of one-many relations, if we 
had not wished to derive them from the more general notion of a — > /S. In 
proving that a relation is one-many, *7l*17 is more often employed than any 
other proposition. 

*71'22. b:Rel^>Cls.S<ZR.3.S6l-*Cls 

*7125. h . R, S e 1 -» Cls . D . R 1 8 e 1 -> Cls 

*71-36. h:.Rel-+Ch.Dzx = R'y. = .xRy 

*71-381. I- : R e Cls -+ 1 . D . R"(a -0) = R"a - R"/3 

(This proposition is more useful than the corresponding property of 
l->Cls.) 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 429 

*71-55. \-i:Rel->Cls.D:.Rtp6l->l. = iy,Z€0.R<y = R t z.'2y, z .y = z 

This proposition is constantly used. For example, putting Q. for R, it 

gives 

h:.ar/3el-+l.= :P,Qe£.(I<P = <3'Q.Dp, <2 .P=Q. 

Most of the relations used to establish correlations in arithmetic are 
obtained from a one-many relation, such as G, by imposing some limitation 
on the converse domain which makes the relation one-one. 
♦71-671. H :.y e .O y .E! R'y :s. R\ /3 el-»Cls . £ Cd'R 

Here "ye/S.Dy.E I R'y" is E !! R"@, which has already played a large 
part as a hypothesis, e.g. in #376 ff. 
*71-7. h :. Qe 1 -* Cls . D : xP \ Qz . = . xP{Q?z) 

Thus for example we shall have x (P | Cnv) R . = . xP (Cnv'JK). 



*71 01. V . 1 -> Cls = R {R"d'R C 1) [*70'4] 

*7102. \-.Cls-*l = R(R"D 7 RCl) [*7041] 

*7103. t-.l-*l = R(R«<I<RCl.R"I)'RCl) [*202 . (*70'01)] 
*7104. r . 1 -* 1 = (1 -> Cls) n (Cls -► 1) [*70-42] 

*711. \-:Rel-+Ck.= .~R"a'RCl [*2033 . *7101] 

*71101. l-:.ReCls-»l. = .jR"D'2SCl [*2033 . *7l02] 

*71102. I- : R e 1 -> 1 . = .^B"a'jB C 1 . £"D<£ C 1 [*20'33 . *71'03] 
*71103. h:i2el->l. = .i2el-^Cls.i?€Cls-»l [*2233 . *7104] 
*7111. H : £ e 1 -> Cls . = .^K"V Clui'A [*7044] 

*71111. r : £ e Cls -> 1 . = . jR"V Clut'A [*70441] 

*71112. h:im^l. = .jR"VClvt'A.S"VClut<A [*70\L2] 

*7112. H:Eel->Cls. = .(2/).i2'y€lut'A t*70'45] 

*71121. \- : R€C\s-+l .= .(x) . R'xel v I'A [*70'451] 

*71122. h:.Rcl^>l.= :(y).R'yelvi'A:(x).R'xelvi,'A [*7013] 
*7113. \-:.Rel-*Cla. = :(y):R'yel.v.R'y = A [*70'46] 

*71131. h : . R e Cls -* 1 . = : (x) : R ( x e 1 . v . E'# = A [*70'461] 

*71132. h :: R e 1 -> 1 . = :. (y) : R'y el . v . R'y=A :. (#) : R l xe 1 . v . R'x=A 

[*7(M4] 

*7114. Ki.lSel-^Cls.^altf'y.iV-R'yel [*7047] 

*71141. t-i.ReCte-tl.zz-.RlR'x.yx.R'xel [*70471] 



430 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

#71142. b :. R el ^1. = -.%! R'y. D y . R'y el :rI R'x. D x .*R'x el [#7015] 

#7115. b : R e 1 -> Cls . = . B'RC 1 u i'A [#7048] 

#71151. Hi2eCls-*l. = .D'jRClut<A [#70481] 

#71152. f- : -B e 1 -> 1 . = . D'.R Clwi'A. D<# C 1 w t'A [#7016] 

#7116. H : R e 1 -* Cls . = . E !! R"<1'R 

Dem. 

b . #37702 . #711 . D 

h :. R e 1 -»Cls . = : y*d'R . D y .~R*y e 1 : 
[#533] = : y e d'R .D y .ElR'y: 

[#37104] = :EllR"d'R:.Db.Tvop 

This proposition is very important; it exhibits the connection of descriptive 
functions with one-many relations. 

#71161. b:ReC\s->l. = . Ell R"T>'R 

#71162. b:Rel-+l. = .Ell R"<1<R . E !! R"T>'R 

#71163. b :. .Re 1 -> Cls . = -.yed'R . = y . E ! R'y 

Dem. 
b . #3343 . D b : E ! R'y . D . y e (F£ : 

[#4-73] D h -..yed'R .O.ElR'y: = -.yed'R . ~. El R'y :. 

[#1011-271. #37104] D h :. E !! R"d'R .= :ye d'R . = v . E ! R'y (1) 

h. (1). #7116. DK Prop 

#71164. I- :. R e Cls -> 1 . = : xe D'R . = x . E I R'x 

#71165. b:.Rel-+l.= -.yed'R . = y . E ! R'y zxeD'R . = x . E ! R'x 

#71166. h:(y).ElR'y.D.Rel-^Ch 

Dem. 

b . #202 . #10-1 . DH:.Hp.D -.yed'R. 3. E! R'y:. 
[*1011-21.*37104] D I- : Hp . D . E !! R"d'R . 
[#71-16] D . i2 e 1 -> Cls : D f- . Prop 

#71167. !-:(*). ElR'x.D.ReCte-+l 

#71-168. b :. (y) . E ! R'y : (x) . E ! R'x : D . £ e 1 -> 1 

*71'17. h :..R el -* Cls. = : xRz .yRz ,"D x , y>z .x = y 

This proposition is constantly used in the sequel. 

Dem. 

— > — * 

K#52-4. D h -..R'zel ui'A. = : x,yeR'z. D x>y .x = y : 

[#32-18] =:xRz.yRz.D Xty .x = y:. 

[*10-11-271.*11-21] D H :. (*) . iZ's e 1 w i'A . s : xRz . yRz . D XtVjZ . x = y (1) 
K(1).*7112. Dh.Prop 



431 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 

#71171. I- :. R e Cls -* 1 . = : xRy . xRz . D x>y)Z .y = z 

#71172. V : . R e 1 -> 1 . = : xRz . yRz .1 Xty , z .x = y. xRy . xRz .^ XiVtZ .y = z 

4— 4— 

#7118. h :. R e 1 -> Cls . = : a ! R'x n R'y . O x<y .x = y 
Dem. 

V . #321 81 . #22-33 . D 



h :. g ! R'x r\ R'y . x>y .x = y: = 
[*10'23] = 

[*7l'l7] = 



(3,2) . xRz . yRz . D Xt y .x = y\ 
xRz . yRz . D Xj y , z . x = y : 
E e 1 -► Cls :.DI-. Prop 



#71181. h :. # e Cls -► 1 . = : a ! E'y n R'z . D y>z .y = z 

#71182. I- :: R e 1->1 . = :. a ! R'x n E'y . v . a ! E'# n R'y : D x>y . x = y 

#7119. h:JKel-»Cls. = .i2!-R = iTD<E 
Dem. 

h . #341 . #31-11 . D V . x (R | R)y . = . (a*) . #i^ . yRz 
\-.*501.*3510l.1\-.x(I\-I)'R)y. = .x = y.yeI)'R 
K (1) . (2) . #21-43 . D 

H::i2| J R = /tD' J R.= :.(a^).«^.y^-^,y :a; = 2/-y eI) ' jB: 
[*3313,*10'35] =*,„ : (a*) .x=y.yRz: 

[#13194] = x , y : (a*) .x = y.xRz.yRz: 

[#10-35] =* ty :a> = y:(a*).atffe.y.Rs:. 



(1) 

(2) 



[#4-71] 

[#10-23] 

[#71-17] 



. (a*) ■ xRz . yRz .0 Xt y.x = y.. 
. xRz ^yRz . ^> x ,y,z .x = y :. 
. R e 1 -* Cls ::DK Prop 



#71191. l-:iJeCls->l. = .i2|J2 = /r a '- B 

#71192. \-:Rel->l. = .R\R = ItD'R.R\R = It<I'R 

#71-2. t- . Cls -» 1 = Cnv"(l -> Cls) . 

1 _» Cls = Cqv"(C1s _► 1) . 1 -► 1 = Cnv"(l -► 1) [*70"22] 

#71-21. h:J2el->Cls. = .JReCls->l 
Dem. 

h . #37-62 . #31-13 . D b : R e 1 -► Cls . D . Cnv'iZ e Cnv"(l -> Cls) . 

[*31-12.*7l-2] D.SeCls-*l (1) 

h . #37-62 . #3113 . D h : £ e Cls -> 1 . D . Cnv'R e Cnv"(Cls -> 1) . 
[*31-33.*7l-2] D. Eel-* Cls (2) 

h . (1) . (2) . D h . Prop 



432 



PROLEGOMENA TO CARDINAL ARITHMETIC 



[PART II 



#71-211. \-:ReCls->l. = .R € l-+Cte 

#71-212. h:Rel-+l. = .Rel^>l 
*7122. \-;Rel-+C\8.S(ZR.D.g€l- 
Bern. 



Cls 



*71221. h 
*71'222. Y 
*71223. h 
*71224. Y 
#71-225. h 
*71*23. Y 
#71-231. h 
*71232. Y 

#71233. Y 
Dern. 



K*23'1.3 

I- :. 8 G R . 3 : xSz . ySz . 3«, tf , z • #-&z • yRz 

Y . #71-17 .3 

h :. i2 e 1 — ♦ Cls . 3 : xRz .yRz . D^j/.z •® = y 
K(l).(2).*ll-37.3 

Y :. Hp . 3 : #$? . ySz . 3 a . >t , >z .x = y: 
[#7117] 3 : S € 1 -> Cls :. 3 I- . Prop 

JKeCls-^l.SGiSO.SeCls-^l 

Rel-*l.S<ZR.D.Sel-*l 

Re 1 ->Cls . 3 . RKRC 1 -»Cls [#71*22 .#61-2] 

EeCls-»1.3.Rl'#CCls-»l 

R e 1 -> 1 . 3 . R1«R C 1 -* 1 

12 e 1 -> Cls . 3 . 12 n £ e 1 -> Cls [#71-22 . #2343] 

.ReCls->1.3.irA£<:Cls-+l 

Rel-+1 .D.RnSel-*l 

R,S€l-+Cte.3.RnSel->l 



(1) 
(2) 



(1) 



(2) 



Y . #71-23 . 3 H : Hp . 3 . R n S e 1 -* Cls 

H . #71-21 . 3 h : Hp . 3 . S e Cls -* 1 . 

[*71231] 3.JSn£eCls-»l 

K (1). (2). #71-103. 3 K Prop 

#71234. \-:R,SeC\8-*l.-}.RnSel^>l 
#71-235. h:126l-»Cls.£eCls-»1.3.12n,Sfel-»l 
#7124. \-:R,Sel-*Cte.<I'Rr*a'S = A.3.RvSel-+Cte 
#71-241. h : R,S eCls-> 1 . D'i2 n D'£ = A . 3 . R vSeC\s-+ 1 

#71-242. I- : R, Se 1 -> 1 . D'iJ n D'£ = A . d'12 n<P£ = A.3.Ec/£el-*l 

[#70-56] 
#71-243. l-:l£,£el->l.C"12nC"£=A. 3. RvSel-^1 [#7057] 

*71-244 K:12,£el^Cls.l2r<P,SfGS.3.12c-£el-»Cls 
Dem. 

Y . #23-34 . #4-4 . 3 

Y :.x(Rw S)z .y(Rw S)z .= :xRz .yRz .v .xRz .ySz .v .xSz .yRz .v .xSz .ySz (1) 



[#70-54] 
[#70-55] 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 433 

h. *71 17 . D h :. R,S el-*C\s .1:xRz .yRz . D . x=y z xSz . ySz.1.x=y (2) 
h . #3314 . #4*7 . D h : xRz . ySz . D . #£* .ySz.zed'S. 

[#35101] D.x(R\><l'S)z.ySz (3) 

K(3). DI-r.iJ^a'SG^f.D^^.^.D.^^.^ (4) 

h.(4)^. Dhi.JR^a^G^.D^&.^.D.^^.^ (5) 

K(2).(4).(5). D h :.B.p .D : xRz . ySz ."D . x = y : xSz . yRz . D .x = y (6) 
I- . (1) . (2) . (6) . #4-77 . D I- :. Hp . D : x (R c; £) * . y (R vy fif) z . D . a = y (7) 
I- . (7) . #1011-21 . #7117 . D I- . Prop 

*71245. \-:R,SeC\s-^l.(I> t S) J \RQS.O.R^SeC\s-*l 
#7125. h:i*,Sel-*Cls.D.2£|£el->Cls 

h . #7117 .Dh:.K-p.D:ySx.zSx.D.y = z: 

[Fact] D : uRy . ySx . vRz . zSx . D .y = z . «Ry . vifo . 

[#1313] D . wity . vRy . 

[#71-17] D.w = t> (1) 

h.(l).*llll-3-54.D 

I- :: Hp . D :. (gy) . wJBy . ySx : (g\2) . wifc . zSx : D . u = v :. 

[*34'1] D:.w(i2|#)a-.v(i<:|£)a-.D.w = v (2) 

h.(2).*7l-17.Dh.Prop 

#71-251. H: J R,£eCls->l.D. J R|£eCls->l 
*71-252. h:i2,£el-»l.D.i2|£el->l 

#7125 maey also be deduced from #70-6, as follows: 
Alternative Dem. o/*71*25. 

h . #53301 . #7112 . D I- : R e 1 -* Cls . D . 22"t<# e 1 u i'A : 
[*52-l] Dh-.Rel^Cls.ael.D. R"a e 1 u t'A : 

[#37-611 1103] Dh:.Rel-»Cls.D.i2'"lClw<A (1) 

h. (1). #706. Dh. Prop 
Similarly #71-251 may be deduced from #7061. 
*7126. h:i2el-*Cls.D.Er7€l->Cls [*7062] 

#71-261. h : 22 e Cls -> 1 . D . £ 1 R e Cls -► 1 [#70-63] 

#7127. h:i2el->Cls.D./3 > | J Bel-*Cls [#35*44 .#71-22] 

#71-271. h:i2eCls-*l.D.Ep7eCls->l 

#7128. Hriiel-^Cls.D.yS^^el-^Cls [*35'442 .#71-22] 
#71-281. h-iZeCls-^l.D.ySli^eCts-*! 

#71-29. h:i2el-^l.D.^1 J R, J Rr7^1- R r7 €l -> 1 

#71-31. I- : R e 1 -» Cls . y e <I<i2 . D . (ify) % [*30'32 . #71163] 

K&W I 28 



434 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*71-311. \-:ReCte->l.xeD'R.D.xR(R'x) 

*71312. H : R e 1 -* 1 . x e D'R .y e d'R . 3 . xR(R'x) . (R'y) Ry 

*71'32. b::R€l-^Cl8.y€a'R.^:.^(R'y). = z(^.x).xRy.^x- = zxRy.^ x .yfrx 
[*30-33.*7M63] 

*71'321. \-i:ReC\a^l.xeJ)'R.>:.1r(R'x). = z(fty).xRy.^i = :xRy.3y.ylry 

*71'33. ViiRel^Q\s.^u^(R'y)i = i(Rx).xRy.^xi = :yea'RixRy.? x .tyx 

Bern. 

K*71-32.*532.D 

h :: Hp . D :. y e Q.'R . yjt (R'y) . s : y e G.'R : (g#) . xRy . i/r# c= 

^zyed'R-.xRy.^.^x (1) 

H . *1#21 . D I- : ^ (2fy) . D . E ! R'y . 
[*33'43] D.yed'R: 

[*471] DI-:.y€(I'.R.^(.R'y). = .'^(.R'y) (2) 

>.*105. D I- : (g#) . xRy . ^r# . 3 . (g#) . a?.% . 
[*33131] D.yeCKR: 

[*4-71] DHi.yeGL'E^a^.ajEy.^rs.^^.ajJSy.^ (3) 
K(l).(2).(3).DKProp 

*71'331. h :: i2eCls-> 1 . D :. f (R'x) . = : (ay) .xRy.fy: = : 

x e D'R : xRy . D y . ^y 

*71'332. I- :. Eel ->Cls . D : R'y ea. = . g ! i2'y r> a . = .yed'R.R'yCa 

*71*333. I- :. i2 eCls -> 1 . 3 : i?'a?ea . = . a ! R'x rs a . = .xeI>'R . R c xC a 

*71'34. h : £ e 1 -* Cls . R = 8 . y e (KB . D . J2'y = S'y [*3036 . *7ll63] 

*71'341. \-:ReC\s^l.R = S.x€D'R.D.R'x = S'x 

*71'35. H:: J B e l->Cls.D:.yea'i2ua'/Sf.D 1 ,.KV=^: = -^ = ' Sf 
Dera. 

h.*2118. D\-:.R = S.D:yea<R»a'S. = .ye(I'Ryja'R. 
[*22-56] =.yea'i2 (1) 

h.(l).*71'34.3h::Hp.E = ^.D:y€a'i2ua t >Sf.D y . J B^ = /Sf'y (2) 
I- . (2) . *3345 .OH. Prop 

*71351. \-::R € Ch^l.D:.xeJ)'Ryj D'S.3 x .R'x = S'x: = . R = S 

*71352. b::R€l^l.D:.yea'Ryja'S.D y .R'y = 8'y: = :R = S: 

= :xeI><Rv*D'S.O x .R'x = S'x 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 4 35 

#7136. h :. Re 1 -> Cls . D :x = R t y.= .xRy 

Bern. 

K*30-4.*7 1163. D 

\-:.HV.ye<I'M.D:x = R'y. = .xRy (1) 

h.*7l!63. Transp. D 

I- : . Hp . y ~ e (F i2 . D . ~ E ! R l y . 

[*14'21 .Transp] D . ~ (# ** .R'y) (2) 

I- . *3314 . Transp . D H : y ~ e d<22 . D . ~ (#%) (3) 

I- . (2) . (3) . #521 . D 

H.Hp.y~e(I'.#.D:a; = J K'y. = .a? J Ry (4) 

K (1) . (4) . *4'83 . D V . Prop 

*71'361. I- :. R e Cls -* 1 . >: y = R'x. = . xRy 

*71362. \-:.Rel-*l.D:x = R'y. = . xRy . = .y = R'x 

*7137. \-:.Rel-+Ch.D:yeR"a. = .R'yea 

Dem. 

h . #71*33 . D h :. Hp . D : R'y e a . = . (ga?) . a?i2y . x e a . 

[#37105] = .ye£"a:.:>KProp 

#71-371. h:.ReCls->l.D:x€R"a. = .R<x€a 

#71-38. h : # € 1 -* Cls . D . 5"(a - /3) = E"a - R"P 
Dem. 

K*7l-37.Dh:.Hp.D:yeE%x-/3). = .i2'yea-/3. 
[*22-32.*14-21] =.R'yea.~(R'yel3). 

[*7137] =.yeR"a.~(yeR"l3). 

[*22-32J = . t/ e .R"a - R"& :. D h . Prop 

*71381. h : R e Cls -* 1 . D . iZ"(a - /3) = i2"a - R"p 

*714. H : i2 e 1 -> Cls . D . i2"/3 = x {(ay) .yefi.x = R'y} [*37l . *71'36] 

#71-401. h : J? e Cls-> 1 . D . 11"$ = $ \{^x) .xe&.y = R<x} 

*71'41. h : 22 e 1 -»Cls.D.D' J R = £ {(ay). a> = .R'y} [#3311 .#71-36] 

#71-411. H : R e Cls -* 1 . D . (F12 = {(a*) . y = R'x) 

#7142. r : : R e 1 -» Cls . /3 C CI<.K . D :. R"$ Qa. = :y e $ .^y.R'yea 
[*37-61 . *7116] 

#71-421. h::R€Cte-+l.aCT>'R.3-..R"ciCj3. = :xea.D x .R'xe{3 
#71-43. h:.Rel-»Cls.yean(F J ft.D. J R<yei2"a [#3762 . #7116] 

#71-431. t-:ReC\s-*l.xear\D f R.3.R<cceR"a. 

28—2 



436 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*71'44. h :: Re 1 -> Cls . a C (Pi? . D :.xeR"a .1 x .^x; = :yea. D y .yfr(R'y) 
[#37-63 . *71\L6] 

*71441. V :: i2eCls-»l . aCB'R . D :. y e R"a.D y .tyy : = : x ea .O x . ^r(R'x) 
#7145. H :. R e 1 -» Cls . D : (g#) . a; e E"a . yfrx . = . (gy) . yea.f {R'y) 

Dem. 
V . *37'64 . #71-16 . D 

h :. Hp . D : (>&x) .xeR"(a n d'R) . yjrx . = . (<&y) .yean d'R .^(R'y) (1) 
h.*37-26. D h . R"(a n d'R) = R"ol (2) 

l-.*14-21. Db :y ea. yjr(R'y). 5. El R'y. 
[*33-43] D.ye d'R : 

[*4-71.*22'33] D h : y e a . -f (R'y) . = .yean d'R . yfr (R'y) : 
[#1011-281] D I- : (gy) . y e a . ^ (JB'y) . = . (gy) .j/ea^'iJ.f (fl'y) (3) 
h.(l).(2).(3).DI-.Prop 

#71451. h:.ReC\s-+l .3 :(^y) .y eR"a.^y . = .(^x) .xea^(R'x) 

#7146. 1- : i? e 1 -* Cls . a C #"/3 . D . a = R"(R"a n 0) 
Dem. 

r . #37-26 . D H : R"/3 = R"(/3rxd'R) . R"(R"anj3) = R"(R"an/3nd'R) (1) 
I- . #37-65 . *71-16 . D 

h : # e 1 -» Cls . a C iZ"(/3 n d'i2) . D . a = R"(R"a n £ n <P#) (2) 

K(l).(2).DKProp 

#71-461. H : 22 e Cls -> 1 . /3 C ii"a . D . /3 = R"(R«/3 n a) 

*71-47. h :. R e 1 -> Cls . D : a C R"/3 . = . (g 7 ) . 7 C /3 . a = #"7 

I- . *7146. #10-24. #22-43 . D I- :. Hp . D : a C R"f3 . D . (37). 7 C /3 . a=B" 7 (1) 
K #37-2. #10-11-23. 3\-:(>&v).yC/3.a=R"y.D.aCR"j3 (2) 

h.(l).(2).Dh.Prop 

*71471. H :. R eCls-» 1 . D : /3 C #"a . = . (37) . 7 C a ./3 = R"y 
#7148. > : i£ e 1 -» Cls . D . D'i? e = CPD'iJ 

Dem. 
r . #37-24 . #60-2 . Dr. D'i2 f C CI 'T>'R (1) 

1- . #37-25 . #71-47 . *60'2 . D I- : Hp . a e C\'T>'R . D . (37) . 7 C <J'i£ . a = R"y . 
[*10-5.*37-23] D.aeD'i2 £ : 

[Exp.*1011-21] DhiHp.D.Cl'D'iZCD'ik (2) 

h . (1) . (2) . D H . Prop 

#71-481. f-: J ReCls-»l.D.D<( J R) 6 = CKF£ 

The following proposition is used in the theory of derivatives of a series 
(#216-411). 



SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 437 

*7149. r : Rel->Ch.aCa'R.3.R"<C\'a=C\<R"a.R"'C\ex<a=C\ex'R"a 

Dem. 
h.*71-47.*60-2.Dh:.Hp.D: 7 eCl^"a. = .(a)S).^Ca.7 = i2")S. 
[#37103] = . 7 ei2"'Cl'a (1) 

h . #3743 . D h :. Hp . /3 e Cl'a . D : 3 ! £ . = . 3 ! R"$ (2) 

h.(l).(2).DKProp 
#71*491. h: J R€Cls^l.aCD < J R.D.5 t "Cl < a=Cl < ^"a.5" < Clex'o=Clex'^"a 

This proposition is used in the theory of derivatives of a series (#216-4) 
and in the theory of ordinal numbers (#251-11). 

#71-5. h :.i2el-*Cls.D :xRy. = . x=u'R'y 
Dem. 

V . #71-36 . #30-1 . D h :. Hp . D : xRy . = .x = (ix) {xRy) . 

[*51-56.*32-13] = . x = I'R'y :. D I- . Prop 

#71-501. h :. ReC\s^>l .3 ixRy .-= .y ^'R'x 

#71-51. h : R e 1 -* Cls . y e d'iS .D.R'y = l'R { y 
Dem. 

h . #5331 . #71163 . D h : Hp . D . t'R'y = B'y . 
[#51-51] D.R'y = ^'R'y : D H . Prop 

#71511. h : -R e Cls -> 1 . x e D'R .D.R'x = \ l R'x 

#71-52. r- : JB e 1 -* Cls . D . R"a = 7"E"a 

Dem. 
h.*37\L. DKt"i£"a = &{(3/3)./3eI2"a.#i/3} 

[#51-51] =^Ka^).y8e J B"a.«=t < )S} 

[#37-7] = d {(^8, y).yea. $ = ~R'y . x =>/3} 

[*11-23.*1 3-195] =a{(ay).yeo.fl?=^'22'y} (1) 

I- . (1) . #71-5 . D h : Hp . D . \"R"* = x {(%y) . 3/ ea . xRy) 
[#37-1] = R"a Oh. Prop 

#71521. h : R e Cls -* 1 . D . X"a ="i'?R"ci 
#71-53. h:Eel-^Cls.^ = J R < 2/- :) - a; = y 

r- . #14-21 . D r- : Hp . D . E ! R'x . E ! R'y . 

[#30-32] D . xR (R'x) . yR (R'y) . 

[#14-16] D . xR (M'y) . yR (R'y) . 

[#71-17] D.a = 2/:DKProp 



438 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II 

*71531. b:ReC\s-+l.R'y = R'z.D.y = z 

*71532. b:.Rel->l.3:R'y = R'z.D.y = z:R'x = R'y.'2.x = y 
*7154. b ::R el ->Cls . D :. i2 e 1 -> 1 . = :R'y = R'z. D ytZ .y = z 

This proposition and the next (#71/55) are very often used. 

Dem. 
b . *71'36 . D h :. Hp . D : (g«) . xRy . xRz . = y , z . (%x) .x = R'y .x = R'z. 
[*14-205] =y tZ .R'y = R'z (1) 

b . (1) . D b :: Hp . D :. R'y = R'z . D y>z .y = z: = : (g#) . xRy . xRz . D y>z . y=z : 
[*10-23] = : xRy . xRz . Xty>z .y = z: 

[*71-171] =:ReC\s-+l (2) 

b . *71103 . *473 . D b :. Hp . D : R e Cls-* 1 . = . R e 1 -* 1 (3) 

H . (2) . (3) . D I- . Prop 

*71-55. \-::Rel-+C\s.D:.Rtl3el^>l. = :y,z€l3.R'y = R'z.Dy tZ .y = z 

Dem. 
b . *71'26 . D h :: Hp . D :. R f /3 e 1 -► Cls :. 

[*71'54] Dr.^r/Sel-^l.s:^^ /8)'y = {R[ j3)'z .O y>z .y = z: 
[*35-7] =:y,zel3.R'y = R'z .0 y , t .y = z::Db. Prop 

*7156. r-:.2Sel^l.ye(KR.D:.R'y = 22's.=E.y = s 

Dem. 

h.*71-532. Db:Up.R'y = R'z.D.y = z (1) 

h.*71165.*30-37.Dl-:Hp.2/ = 2.D.i2 f 2/=E^ (2) 

K(l).(2).DKProp 

*71561. b:.Rel-*l.xeD'R.D:R'x = R'y. = .x = y 
*71-57. b :. R'y = R'z .= y>z .y = z : = : R el ^>1 : (y) .El R'y 

Dem. 

b . *10-1 .Ob:.R'y = R'z.= y>z .y = z:D:R'y = R'y.= y .y = y: 

[*1315] D:(y).R'y = R'y: 

[*1428] D:(y).ElR'y (1) 

[*71166] D:Rel-*C\s (2) 

I- . (2) . D r :. Hp (2) . D : R e 1 -» Cls : R'y = R'z .D y , z .y = z : 

[*71'54] D:Rel-*l (3) 

l-.(l).(3).*7l-56. Dh.Prop 

*71571. b :. y e £ . D y . E ! R'y : = . R \- /3 e 1 -> Cls . /3 C a'JB 

Z)ew. 
h. *71-16. D h :. R\-/3e 1 -> Cls. = : y ed'(Rt /3) .D y .E l(R\-/3)'y: 
[*35-64'7] =:ye/3na'R.D y .ye0.ElR'y: 

[*22'33.*5-3] =:ye/3n d'R . D y . E ! R'y (1) 



SECTION C] ONE-MANY> MANY-ONE, AND ONE-ONE RELATIONS 



459 



K(l). #22-621. Z> 

\-:.Rt/3el-*Cte.!3Ca<R. = :yel3r>a<R.Oy.ElR'y.j3*a'R = P: 

[•18193] = :ye/3.D y .ElR'y:f3«a<R = l3 (2) 

H.*33-43\DI-:.ye/8.D y .B!i2'y:D./8Ca'i2. 

[*22'621] D.0na'R-0 (3) 

K(2).(3).*4'7l.DKProp 

#71-572. \-:.yc0na'R.'Dy.'&lR i y. = .RtPel-*Cia 
[#71-571 . #35-351 . #22-43] 

#71-68.' \-ny,gefi.D v , s iR t y = R t z. = .y = zz.D.R[fi€l^l.fiCa t R 

Dem. 

\-.*101.0\-::H.v.3:.yel3.Dy:R'y = R i y.= .y = y: 

[*13-15.*14'28] lyiElR'ys. 

[#71-571] D:..R| k £el-*Cl8.0C<rJR (1) 

h . #3-26 . Imp . #1111-32 . D 

I- :.Hp.D:y,**e£..B'y = £'*.:>„,,.? = *: 

[#35-7] 0:(R^yy = (R[^) ( z^ y>z .y = zz 

[#71-54.(1)] D:Rt/3el-+l (2) 

K(l).(2).DKProp 

#71-59. h::y^6/3.D y>z :i2 < y = i2^.= .2/ = ^:. = .-Rr/ 9el -» 1 -^ C(Ifi2 J 

Dem. 
H. #71-56. DH::i2f-^el-»l. D:.yea'(i2 1^/3). D:(i2Wy=(-RW^. = .y=^:. 
[#35-64-7] ^i.ye^^a'R.^:y y z^.R l y^R l z.=.y^z (1) 

h. (1). #22-621. Dh::J2r/8 e l->l./3Ca'E.D:. 

2/ e £ . D : y, z e £ . R*y = -R's . = . y — z :. 
[#4-73] D:.y y zep.D:R'y = R l z. = .y = z (2) 

F.(2).*llir3.Dl-:: J Rr/3el->l./3Ca < J R.D:. 

y, s 6/3 . D,,, : R l y = R t z.= .y = z (3) 

K (3). #7 1-58. DK Prop 

The following proposition is used in the theory of selections (#80"91). 

#71-6. - \-:R e l-*C\8.0.R = s<P\(ny).ye<I<R.P = (R<y)ly} 

Dem. 

\- . #41-11 . #13195 . D 

\-:x[s t P{(' 3 _y).yea t R.P = (R t y)iy}liz. = . 

(< K y).ye<I<R.x{(R<y)ly}z. 

[#5513] =.(ny).yea<R.x = R&l